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NTH ROOTS OF MATRICES
by
Eric Jarman
An Abstract
presented in partial fulllment
of the requirements for the degree of
Master of Science
in the Department of Mathematics and Computer Science
University of Central Missouri
June, 2012
ABSTRACT
by
Eric Jarman
This paper investigates the feasibility of nding any
power, for an arbitrary
m×m
m × m square matrix.
matrices and complex
m×m
nth root,
and by extension any rational
Root nding methods are investigated for real
matrices.
NTH ROOTS OF MATRICES
by
Eric Jarman
A Thesis
presented in partial fulllment
of the requirements for the degree of
Master of Science
in the Department of Mathematics and Computer Science
University of Central Missouri
June, 2012
iv
NTH ROOTS OF MATRICES
by
Eric Jarman
ACCEPTED:
C~o:::hematics
and Computer Science
Contents
1
Introduction
1
2
Iterative Methods
3
2.1
Babylonian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Binomial Theorem
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.3
Netwon's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.4
Denman-Beavers Iteration
2.5
Weaknesses
3
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Complex Number Equivalents
13
3.1
Complex Numbers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
3.2
Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3.3
Cayley-Dickson construction . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
Matrix Forms and Properties
22
4.1
Powers of a Matrix and Diagonal Matrices
. . . . . . . . . . . . . . . . . . .
22
4.2
Matrix Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4.3
Examples: Diagonalizing Matrix Forms of Complex Numbers . . . . . . . . .
29
4.3.1
Example 1:
2×2
Matrix of a Complex Number . . . . . . . . . . . .
29
4.3.2
Example 2:
2×2
Matrix of a Quaternion . . . . . . . . . . . . . . . .
34
v
vi
Contents
4.3.3
5
4×4
Matrix of a Quaternion . . . . . . . . . . . . . . . .
38
4.4
Roots of Complex Numbers vs. Roots of Diagonal Matrices . . . . . . . . . .
42
4.5
Jordan Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.6
Nilpotent Matrices
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.7
Existence of Roots
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
Hypercomplex Numbers
55
5.1
Multicomplex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.1.1
Bicomplex Numbers
. . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.1.2
Tessarines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.2
Split-Complex Numbers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
5.3
Dual Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.4
Split-Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
5.4.1
Roots of Split-Quaternions . . . . . . . . . . . . . . . . . . . . . . . .
62
Biquaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
5.5.1
68
5.5
6
Example 3:
Conclusion
Bibliography
Roots of Biquaternions . . . . . . . . . . . . . . . . . . . . . . . . . .
72
73
Chapter 1
Introduction
The operations for nding any integer power of a matrix are well-dened. The same, however,
cannot be said for non-integer powers of a matrix. If we can nd a method for calculating
some root of a matrix
integer
n > 1,
A,
that is to say, nd some matrix
then we can dene this matrix
A0
to be an
It is important to note that for an arbitrary matrix
of
A.
A0
nth
A,
where
root of
(A0 )n = A
for some
A.
there may be multiple
nth
roots
It is easy to construct a simple example which shows this:

2 
2 

4 0
−2 0 
2 0
.
 =

 =
0 4
0 −2
0 2
When there are multiple solutions, we can choose one of them as the principal root. If we
do so, we can denote the principal root by
1
An .
We should note that it is possible to construct examples of complex matrix roots as well
as real matrix roots:
2 
2 

 i 0
−i 0 
−1 0 

 =
 =
.
0 i
0 −i
0 −1

So, we would ideally like to have root nding methods which will work for real matrices and
1
CHAPTER 1.
2
INTRODUCTION
complex matrices.
This denition can also be used to nd any rational power
the principal
nth
root to the
pth
p
of a matrix by simply taking
n
power
A
1
n
p
p
= An .
We note that this construction only has meaning when
matrices. There is no need to examine
m×n
construct matrices whose product is a given
matrices where
m×n
A
and
m 6= n,
A0
are square
m×m
as we might of course
matrix, but such matrices would not be
equal in values or dimensions, and so cannot be called roots.
Chapter 2
Iterative Methods
We can begin by searching for methods of directly calculating a root. The simplest place to
start is to look at methods of calculating square roots of real numbers, and expand them to
be used on matrices.
2.1
Babylonian Method
The Babylonian method, also known as Heron's method, is based on a historical method
of calculating square roots.
The earliest reference to this method is listed in a numerical
approximation for the square root of 2 on the Babylonian clay tablet YBC 7289 [20]. The
rst explicit denition of the method was given by Hero of Alexandria in the rst century
[1].
This method for nding the square root of
s ∈ R starts with
root, and can be fully described as follows:
√
s,
1
s
=
xn +
2
xn
x0 ≈
xn+1
3
for
n > 0.
some approximation for the
CHAPTER 2.
4
ITERATIVE METHODS
This sequence is guaranteed to converge quadratically to the principal root if both
s
and
x0
are positive.
This method can be extended to matrices by starting with the identity matrix
I,
and
proceeding as follows:
X0 = I
Xk+1 =
1
Xk + AXk−1 .
2
This method is numerically unstable, i.e., small perturbations of the error matrix in the
k th
iteration may induce perturbations with increasing norm in successive iterations, causing the
sequence to diverge from the root [24]. When it does converge, this sequence will converge
quadratically to the root. The computational expense of this method is low, requiring the
calculation of a single matrix inverse per iteration.
Note that the norm in the preceding paragraph is the norm of a square matrix.
norm of
A,
denoted
N (A)
or
kAk,
is a nonnegative number associated with
A
The
having the
properties [50]
1.
kAk > 0
2.
kkAk = |k|kAk
3.
kA + Bk ≤ kAk + kBk,
4.
kABk ≤ kAkkBk.
when
A 6= 0,
and
kAk = 0 ⇔ A = 0,
for any scalar
k,
There are multiple methods which may be used to compute the norm of a matrix
of the simplest to compute norms [46] are
NI (B) =
sX X
i
j
b2ij
B.
Two
CHAPTER 2.
5
ITERATIVE METHODS
and
NII (B) = max
X
i
2.2
|bij | .
j
Binomial Theorem
The binomial theorem can be used to nd the coecients of any power of the quantity
For
x, y ∈ R, n ∈ Z, n ≥ 0,
(x+y).
we state:
n n X
n n−k k X n k n−k
(x + y) =
x y =
x y .
k
k
k=0
k=0
n
The well-known generalization of the binomial theorem by Sir Isaac Newton allows for any
real exponent, rather than being limited to positive integers, by replacing the nite sum with
an innite series.
∞ X
r r−k k
(x + y) =
x y .
k
k=0
r
In this case, it is necessary to dene what is meant by a binomial coecient with any real
upper index. If
n
is an integer we have
n
n!
=
k
k!(n − k)!
For any real number
r,
we factor out
(n − k)!
the usual binomial coecient, and substitute
r
from the top and bottom of the formula for
for
n
to yield [21]
r
r(r − 1) · · · (r − k + 1)
(r)k
=
=
,
k
k!
k!
where
(r)k
denotes the falling factorial, dened as
To nd an arbitrary root of any real
a,
(r)k = r(r − 1) · · · (r − k + 1).
we substitute
a = 1+x
into the generalized
CHAPTER 2.
6
ITERATIVE METHODS
binomial equation above to yield a formula involving only a single variable.
(1 + x)
r
=
∞ X
r
k
k=0
xk
= 1 + rx +
r(r − 1) 2 r(r − 1)(r − 2) 3
x +
x + ··· .
2!
3!
Note that we can use the rst few terms of this series as an approximation for a root of
This method can be extended for nding a root of a real matrix
Rm×m
where
A=I +B
(I + B)
r
=
∞ X
r
k
r(r − 1) 2 r(r − 1)(r − 2) 3
B +
B + ··· .
2!
3!
This series will converge if it can be shown that
of
B
as
A, B ∈
Bk
= I + rB +
B k → [0]
[46] by taking
and rewriting the power series as
k=0
show that
A
a.
k
B k → [0]
as
k
increases. It is possible to
increases by either showing that the dominant characteristic root
is less than unity, or by showing that the norm of the matrix
If the norm of the matrix
B
is close to
1,
N (B) < 1.
the series may converge slowly, in which case
the norm of the matrix in the series can be reduced by introducing a scalar coecient. For
A, C ∈ Rm×m ,
and a scalar
k ∈ R,
we take
A = k(I + C),
where
C=
1
k
A−I
to make the
series become
r
A =k
where
k
r
r(r − 1) 2 r(r − 1)(r − 2) 3
I + rC +
C +
C + ··· ,
2!
3!
is any scalar value selected that minimizes the norm of the matrix
C.
The norm
CHAPTER 2.
NI (C)
7
ITERATIVE METHODS
is minimized [46] by using
P P 2
i
j aij
k= P
.
i aii
Other norms may be more dicult to minimize.
The calculation of this series is even cheaper than the Babylonian square root method
shown earlier, since only one matrix multiplication is required for each term, rather than
one matrix inversion. If the series can be shown to converge, or modied appropriately to
ensure faster convergence, only the rst few terms of the series are necessary to nd an
approximation of a root. Also, since we may choose any real value for
r,
we are not limited
to nding only square roots.
2.3
Netwon's Method
Newton's method, or the Newton-Raphson method, is a method for nding roots (or zeroes)
of real valued functions. The method starts with an initial guess
using the function
f
and its derivative
f (xn )
f 0 (xn )
This method can be used to nd a square root of a number
and its derivative
f0
and proceeds to iterate
f 0:
xn+1 = xn −
f
x0 ,
as:
f (x) = x2 − a
f 0 (x) = 2x
a by simply dening the function
CHAPTER 2.
8
ITERATIVE METHODS
We should note that given this denition for the real function
f (x),
we see that the Baby-
lonian method is just a special case of Newton's Method for the square root.
f (xn )
f 0 (xn )
x2 − a
xn − n
2xn
2
2xn − x2n − a
2x
2 n 1 xn − a
2
xn
1
a
xn −
.
2
xn
xn+1 = xn −
=
=
=
=
The proof of quadratic convergence [51] shows us the properties of successive error terms of
the sequence
{xn },
n+1 ≈
f 00 (α) 2
2f 0 (α) k
from which it is possible to determine conditions on
x0
that guarantee convergence of the
sequence:
Let I be the interval
1.
f 0 (x) 6= 0; ∀x ∈ I
2.
f 00 (x)
3.
x0
Then
[α − r, α + r]
r ≥ |α − x0 |
where
,
is not unbounded,
is suciently close to
{xn }
for some
∀x ∈ I
α
,
.
will converge to a square root
α
of
a.
Since the function used for nding a square root of a number
see that the rst derivative
f 00 (x) = 2
means that
f 0 (x) = 2x
will only equal
is constant, and thus not unbounded.
x0
0
at
x = 0,
a
is
f (x) = x2 − a,
we can
and the second derivative
For the initial guess, suciently close
is in a neighborhood of a root within which the sequence can be guaranteed
to converge. One set of conditions for nding such a neighborhood is found in Kantorovich's
Theorem [25]:
CHAPTER 2.
9
ITERATIVE METHODS
Let a0 be a point in Rn , U an open neighbor-
Theorem 2.3.1 (Kantorovich).
hood of a0 in Rn , and f~ : U 7→ Rn a dierentiable mapping, with its derivative
h
i
Df~(a0 ) invertible. Dene
h
i−1
~h0 = − Df~(a0 )
f~(a0 ), a1 = a0 + ~h0 , U0 = B|~h0 | (a1 )
h
i
~
If U0 ⊂ U and the derivative Df (x) satises the Lipschitz condition
h
i h
i
~
~
Df (u1 ) − Df (u2 ) ≤ M |u1 − u2 |
for all points u1 , u2 ∈ U0 ,
and if the inequality
h
i−1 2
1
~
~
f (a0 ) Df (a0 ) M ≤
2
is satised, the equation f~(x) = ~0 has a unique solution in the closed ball U0 , and
Newton's method with initial guess a0 converges to it.
Note that
M
above is the Lipschitz ratio, measuring the change in the derivative.
By modifying our initial function to
nd higher
nth
f (x) = xn − a,
we can use Newton's method to
roots as well. In this general case, the convergence criteria on the rst and
second derivatives are met by any interval which does not contain
use Kantorovich's Theorem to nd an initial guess
x0 .
x = 0,
and we again
The convergence criteria can also be
adjusted to apply to complex valued functions [26], meaning that this method is not restricted
to nding only real roots. Since we see some advantages to this method, we should continue
investigating this method as it applies to matrices.
Newton's method can be extended to matrices by starting with some initial guess
X0
for
a root, and expanding the iteration using Fréchet derivatives.
Fréchet derivatives are used to nd the derivative of a function at a point within the
domain of the function. A simple denition is: A function
f
is Fréchet dierentiable at
a
if
CHAPTER 2.
10
ITERATIVE METHODS
the limit
f (x) − f (a)
x→a
x−a
lim
exists [47]. When applied to a matrix function, for example
F : Cn×n → Cn×n , it is necessary
to remember that matrix multiplication is not commutative for all matrices.
working with matrix derivatives, for any
A, B ∈ Cn×n
Thus, when
we may have something like [19]
d(AB) = d(A)B + Ad(B).
Now to apply Newton's Method to nd a square root of a matrix, we start with
Cn×n
and a matrix function
F : Cn×n → Cn×n
X, A ∈
dened as
F (X) = X 2 − A
To dene the iteration, we use a general function
X0
G : Cn×n → Cn×n ,
taking
as an initial approximation, we have:
−1
Xk+1 = Xk − [G0 (Xk )]
where
G0
is the Fréchet derivative of
G.
G(Xk ),
k = 0, 1, 2, . . . ,
Next identifying
F (X + H) = X 2 − A + (XH + HX) + H 2
with the Taylor series for F,
F (U ) = F (X) + F 0 (X)(U − X) +
F 00 (X)
(U − X)2 + · · · ,
2
F (X + H) = F (X) + F 0 (X)H +
F 00 (X) 2
H + ··· ,
2
Xk ∈ Cn×n ,
and
CHAPTER 2.
11
ITERATIVE METHODS
so
F (X) = X 2 − A,
F 0 (X)H = XH + HX,
F 00 (X) 2
H = H2
2
and so
F 0 (X)
is a linear operator,
so
F 00 = 2,
F 0 : Cn×n → Cn×n .
Thus the iteration can be written as [24]:
X0 given,
Xk+1
=
Xk + Hk
k = 0, 1, 2, . . . .
where
Hk
is the solution of the equation
Xk Hk + Hk Xk = A − Xk2 ,
which can be calculated using the Schur decomposition of
If
F 0 (X)
is nonsingular and
k X − X0 k
converge quadratically to a square root
X
Xk
[4].
is suciently small, the Newton iteration will
of the original matrix
A
[24]. It is important to
note that because this iteration as dened above requires calculating a Schur decomposition
at each step, it is computationally expensive, especially as compared to other iterations.
CHAPTER 2.
2.4
12
ITERATIVE METHODS
Denman-Beavers Iteration
The Denman-Beavers iteration [16] starts with both the identity matrix
I
and the matrix
A,
and iterates in two directions towards a root:
Y0 = A,
Z0 = I,
1
(Yk + Zk−1 ),
2
1
=
(Zk + Yk−1 ).
2
Yk+1 =
Zk+1
This iteration converges quadratically, with
A,
and
Zk
converging to its inverse
X −1 .
Yk
converging to a square root
X
of matrix
The iteration is not guaranteed to converge, even
if a root exists. The computation is relatively expensive, requiring calculating two matrix
inverses at each iteration.
It should be noted that this iteration can be shown to be a
simplication of the general form of the Newton's Method iteration [24].
2.5
Weaknesses
There are various methods which can be used to improve the likelihood of producing a root,
but each of these methods and others like them share a few weaknesses.
numerically based, they all calculate an approximated value.
First, all being
Second, they can all fail to
converge to a solution, even if a root of a given matrix is known to exist. Lastly, they each
only nd a single result, while there may be multiple roots for a given matrix. Given these
limitations, it would desirable to nd some method that would always produce a root, or
further, multiple roots, if such roots exist.
Chapter 3
Complex Number Equivalents
In order to nd a method assured of nding a root of any matrix, it makes sense to rst
look at number systems where roots will be guaranteed to exist and nd matrix equivalents
of these.
3.1
Complex Numbers
The rst number system to investigate is the complex numbers.
To nd any root in the
complex number system, it is necessary only to use a generalization of de Moivre's formula.
Given a complex number
z
in polar form
z = r(cos θ + i sin θ)
or in exponential form
z = reiθ
13
CHAPTER 3.
an
nth
root
α
14
COMPLEX NUMBER EQUIVALENTS
can be written
α = r
1
n
cos
θ + 2kπ
n
+ i sin
θ + 2kπ
n
,
or
1
α = r n ei
where k is an integer with values from
z
n−1
0+i
ei 2
0−i
ei 2
1
2
+
1
2
−
√
i 3
2
√
i 3
2
−1 + 0i
,
providing the unique roots of
eiπ
1
1
to
−1 + 0i
z2
z3
0
θ+2kπ
n
π
3π
π
ei 3
2π
ei 3
eiπ
We can also of course nd the roots of the complex conjugate
z̄ = re−iθ .
z̄ = r(cos x − i sin x)
or
Quickly looking at a few sets of examples of the roots of a complex number
and the roots of its conjugate
values.
z.
1
z̄ n ,
we notice that there do not appear to be any overlapping
CHAPTER 3.
z
ei 2
1+i
√
2
ei 4
√
− 1+i
2
ei 4
1
1
π
0+i
z2
z3
COMPLEX NUMBER EQUIVALENTS
√
i
2
+
i
2
−
3
2
√
3
2
0−i
z̄
π
1
z̄ 2
3π
π
1
ei 6
z̄ 3
5π
ei 6
π
e−i 2
π
0−i
e−i 2
1−i
√
2
e−i 4
1−i
√
2
e−i 4
− 2i +
− 2i −
π
3π
√
3
2
√
3
2
0+i
π
e−i 6
5π
e−i 6
π
ei 2
15
CHAPTER 3.
1
1
z3
π
1+i
√
2
ei 4
+ i sin π8
− cos π8 − i sin π8
√
√3
3
1
1
2
2
+ 2√2 + i 2 − 2√2
2
√
√3
3
1
1
− 2 2 + 2√2 + i − 2 2 − 2√2
ei 8
z
z2
cos
π
8
1
1
z̄ 2
9π
ei 8
π
1
ei 12
z̄ 3
9π
ei 12
e
π
π
π
+ i sin 16
π
π
−i cos 16
+ sin 16
π
π
− cos 16
− i sin 16
π
π
i cos 16
− sin 16
cos
π
e−i 4
− i sin π8
− cos π8 + i sin π8
√
√3
3
1
1
2
2
+ 2√2 + i − 2 + 2√2
2
e−i 8
√
− 1+i
2
√
√3
3
− 2 2 + 2√1 2 + i 2 2 +
e−i 12
cos
1
ei 16
16
z̄ 4
9π
ei 16
17π
ei 16
25π
ei 16
We would like to prove that the roots of
z
π
8
π
16
and the roots of its conjugate
0 ≤ k < n, 0 ≤ l < n ,
θ+2πk
n
θ + 2πk
n
= ei
=
−θ+2πl
n
,
−θ + 2πl
+ 2πm,
n
θ + 2πk = −θ + 2πl + 2πmn,
θ = π(l − k + mn),
⇒ eiθ ∈ R.
z̄
π
9π
e−i 8
π
e−i 12
9π
1
√
2 2
π
− i sin 16
π
π
−i cos 16
− sin 16
π
π
− cos 16
+ i sin 16
π
π
i cos 16
+ sin 16
cos
equal. We can start by seeing what would happen if they were equal. Let
ei
π
1−i
√
2
z̄
i 17π
12
√
− 1−i
2
z4
16
COMPLEX NUMBER EQUIVALENTS
17π
e−i 12
π
e−i 16
9π
e−i 16
17π
e−i 16
25π
e−i 16
will never be
k, l, n ∈ Z, n ≥ 1,
CHAPTER 3.
In other words, the roots of
have
n
distinct
17
COMPLEX NUMBER EQUIVALENTS
nth
z
and
z̄
are only equal when
z = z̄ .
Otherwise,
z
and
z̄
will each
roots. This will be notable later in section 4.4.
We would like to nd a ring isomorphism
Searching yields the mapping
φ from the complex numbers to a set of matrices.
φ : C → R2×2
for a complex number to a real matrix [37]:
z = a + bi,


 a b
φ(z) 7→ Z = 
,
−b a
if
then
where
z
z ∈ C, Z ∈ R2×2 , and a, b ∈ R.
We can then see that the roots of the complex number
can be used to nd roots for the real matrix
Z.
This method has multiple advantages for the matrices in
φ:
it will nd exact roots,
while the iterative methods all nd numerical approximations; it is very ecient, as roots
of complex numbers are easy to calculate; and it will nd
n
roots, namely those which
correspond to roots of the corresponding complex numbers.
The weakness of this approach is that is it not clear that all possible roots for a matrix
in
φ
will be found using this mapping. Further, this mapping cannot be used to represent
any arbitrary
2×2
real matrix, leaving much of the matrix space of the
2×2
real matrices
untouched. Thus, it appears to be necessary to try a larger complex space.
3.2
Quaternions
A larger well-known complex number system is the set of quaternions.
We also have a
method for determining any root of a quaternion, using a variant of de Moivre's formula.
This is achieved by taking a quaternion
z ∈ H, z = a + bi + cj + dk ,
z = a + ω , where a is the real part of z , and ω
If we now take
q
to be a unit quaternion (ie.
and writing it as
is what is called the pure quaternion part of
|q| = 1),
z.
then we can write the unit quaternion
CHAPTER 3.
18
COMPLEX NUMBER EQUIVALENTS
q = a + bi + cj + dk
as:
q = cos θ + ω sin θ,
where
cos θ = a
and
1
(bi + cj + dk)
b2 + c 2 + d 2
1
(bi + cj + dk),
= √
1 − a2
ω = √
a form similar to the polar form of a complex number.
Moivre's formula for a unit quaternion
q
It is then possible to express de
as [8]
q n = eωnθ
= (cos θ + ω sin θ)n
= cos nθ + ω sin nθ.
A generalization of de Moivre's formula shows that
nth
roots of a unit quaternion can
be written [31]
α = cos
where
θ + 2kπ
n
+ ω sin
θ + 2kπ
n
,
k, n ∈ Z, 0 ≤ k < n.
The quaternions are also representable in matrix form, as either a complex
2×2
matrix,
CHAPTER 3.
or a real
4×4
19
COMPLEX NUMBER EQUIVALENTS
matrix, with isomorphisms
φ1 : H → C2×2
and
φ2 : H → R4×4
[52]:
a, b, c, d ∈ R,
if
z ∈ H = a + bi + cj + dk,


 a + bi c + di
2×2
φ1 (z) 7→ Z1 = 
∈C ,
−c + di a − bi

φ2 (z) 7→ Z2

a
b
c
d




 −b a −d c 


= 
 ∈ R4×4 .
 −c d
a −b




−d −c b
a
Even though we can use de Moivre's formula and the isomorphisms above to nd roots
of matrices of the appropriate forms here, it should be noted that in neither of the matrix
equivalences above are we able to represent any arbitrary matrix. We can say that we would
like to nd is some isomorphism which forms a basis for the matrix space.
3.3
Cayley-Dickson construction
We can continue looking at larger complex number elds through the same type of method
used to yield the Quaternion space, namely the Cayley-Dickson construction [18].
This
construction produces increasingly larger complex number spaces, doubling the dimension
each time the construction is used on the previously resulting complex space. In this way,
CHAPTER 3.
20
COMPLEX NUMBER EQUIVALENTS
the complex numbers, quaternions, and larger spaces can be realized as constructions on the
eld of real numbers.
We start by looking at a complex number as an ordered pair
(a, b).
On such ordered
pairs, we dene addition, multiplication, and the conjugate (or involution) as:
(a, b) + (c, d) = (a + c, b + d),
(a, b)(c, d) = (ac − db, ad + cb),
(a, b)∗ = (a, −b).
We take this to be a real algebra
linear spaces
A0 = A ⊕ A,
A,
and construct a new algebra
A0
as a direct sum of
with multiplication dened as
(a, b)(c, d) = (ac − db∗ , a∗ d + cb)
and conjugation extended to
(a, b)∗ = (a∗ , −b).
This construction can now be used to build an innite number of extensions to the complex numbers, known as Cayley-Dickson algebras, each with dimension twice the previous.
However, with each further step, some properties of the algebra may be lost. Applying this
construction to the real numbers yields the complex numbers. When applied to the complex
numbers, we get the Quaternions, which are not commutative. Applied to the Quaternions
this construction results in the Octonions, which are not associative [3]. Applying it further
to the Octonions yields the Sedonions, which are not alternative [27].
Further iterations
CHAPTER 3.
COMPLEX NUMBER EQUIVALENTS
21
from the Sedonions do not lose any additional properties, and remain nicely normed and are
power associative [2].
Given the loss of algebraic properties in the Cayley-Dickson construction, it is important
to note in particular that the Octonions and beyond are no longer associative, meaning that
it will probably not be possible to nd a direct matrix representation for them, as a mapping
from a matrix space would be to an associative subgroup within the Octonions or above.
This leaves only the complex numbers and quaternions with matrix representations known
at this time.
It appears necessary at this point to investigate more properties of matrix
spaces in order to nd some alternate number system in which we might nd an equivalence
or isomorphism to help us search for roots.
Chapter 4
Matrix Forms and Properties
It is not obvious that the iterative or series methods examined earlier are the only methods
for looking for a root of a matrix.
Nor is it obvious that the complex numbers and the
quaternions are the only types of numbers that can be represented in a matrix space. Next
we examine some transformations on matrices and dierent representations of a matrix, and
the properties that can be found or calculations that can be made within each representation.
4.1
Powers of a Matrix and Diagonal Matrices
A power of a matrix is well known, and has the same meaning with respect to matrix
multiplication as a power of an element of a eld such as the real or complex numbers.
When we take any positive integer power
by itself
n−1
n of any matrix A, we simply multiply that matrix
times.
An = A
| ·A·A
{z· · · · · A}
n
22
CHAPTER 4.
When we consider any diagonal real matrix
Cm×m ,
23
MATRIX FORMS AND PROPERTIES
A ∈ Rm×m
or diagonal complex matrix
A ∈
multiplying it by itself looks like this:
A2 = A
 ·A
 a1 0

 0 a2

= .
 ..


0 0

2
a1 0

 0 a2
2

= .
 ..


0 0
···
···
..
.
···
···
···
..
.
···
 

0  a1 0 · · · 0 
 

 0 a2 · · · 0 
0
 

·

.   .
. 
.
.
.
..
.
.   .
. 
 

am
0 0 · · · am

0

0

.
. 
.
. 

2
am
By induction, we can see that to nd any positive integer power
n of the matrix A, we simply
have to raise each element on the diagonal to the same power.
An
We also note that the
0th


n
a1 0 · · · 0 


 0 an · · · 0 
2


= .
.
. 
..
 ..
.
.
. 



n
0 0 · · · am
power of a matrix
A
is dened to be the identity matrix
I.
It
CHAPTER 4.
24
MATRIX FORMS AND PROPERTIES
is clear that this extends the previous denition.
A0

0
a1

0

= .
 ..


0

1

0

= .
 ..


0
0
···

0

a02 · · · 0 


. 
..
.
.
. 

0
0 · · · am

0 · · · 0

1 · · · 0


.
..
.
.
.

0 ··· 1
= I.
When considering a negative power of a matrix, we take such an exponent to be a power
of the matrix inverse of
A.
The matrix inverse of a diagonal matrix yields another diagonal
matrix containing the multiplicative inverse of each value on the diagonal of the original
CHAPTER 4.
25
MATRIX FORMS AND PROPERTIES
matrix in order. Thus, for any negative integer power
n
A−1

a1 0


  0 a2
= 

.
.

.



0 0

−1
0
a1

 0 a−1
2

=  .
 ..


0
0

−n
0
 a1

 0 a−n
2

=  .
 ..


0
0
−n,
A−n =
Thus, we can say that for any diagonal matrix
diagonal to the
nth
···
···
..
.
···
A,
···
···
..
.
···
···
···
..
.
···
−1 n
0 
 

0
 
 
.  
.
.  
 

am
n
0 

0 


. 
.
. 

−1
am

0 

0 

.
. 
.
. 

−n
am
nding
power holds for any integer value of
An
by taking each element on the
n.
It can also be shown that the same is true for any rational power of a diagonal matrix.
CHAPTER 4.
26
MATRIX FORMS AND PROPERTIES
We start by looking at a diagonal matrix
A0
where

0 n
(a1 )

 0

=  .
 ..


0

a1 0

 0 a2

= .
 ..


0 0
0 n
(A )
···
0
(a02 )n
0
···
···
..
(A0 )n = A.
.
···

0 

···
0 


.
..

.
.
.


0 n
· · · (am )

0

0

.
. 
.
. 

am
Taking the entries along the diagonal piecewise, we see
ai = (a0i )n ,
i = 1, . . . , m
Since each element in the arrays is either a real or a complex number (i.e.
can say that
a0i
is an
nth
root of
ai .
matrix
A0 .
1/n
ai
we
We then can choose
a0i = (ai )1/n ,
where we dene
ai , a0i ∈ C),
as the principal
nth
i = 1, . . . , m
root of
ai .
This produces one example of such a
CHAPTER 4.
27
MATRIX FORMS AND PROPERTIES
Now, we can rewrite
A0
in terms of values from

0
a1 0

 0 a0
2

= .
 ..


0 0

1/n
(a1 )

 0

=  .
 ..


0
A0
Given this representation of
A

···
0

··· 0 


. 
..
.
.
. 

0
· · · am
···
0
(a2 )1/n · · ·
..
0
0
.
.
.
.
· · · (am )1/n
0
then
ai
has
n




.



A0 , it is reasonable to dene A1/n = A0 .
of something we might call the principal root of a matrix.
ai 6= 0,

distinct
nth
roots, the matrix
A
This is our rst example
Note that since, in general, if
may have up to
nm
distinct
of this type. We also note that it is not obvious that there are no other roots of
nth
A.
roots
Thus,
we can see that if a matrix is diagonal, this implies that we can always nd roots of that
matrix. In fact, this is also true if the matrix is diagonalizable, as we see in the next section.
4.2
Matrix Diagonalization
To diagonalize a real matrix, we need to nd an invertible matrix
a diagonal matrix [44]. If an
m×m
matrix has exactly
m
P
such that
P −1 AP
is
distinct eigenvalues, then that
matrix is diagonalizable. Note however that the converse may not be true. Thus, if a matrix
does not have
m
distinct eigenvalues, it may still be diagonalizable.
In order to know if
a given matrix is diagonalizable or not, the diagonalization theorem states that an
matrix
If
A
A
is diagonalizable if and only if
is diagonalizable and the
m
A
has
m
m×m
linearly independent eigenvectors [5].
eigenvalues of the matrix, counting multiplicity, are
CHAPTER 4.
λ1 , λ2 , . . . , λm ,
28
MATRIX FORMS AND PROPERTIES
then we can write


λ1 0 · · · 0 


 0 λ2 · · · 0 


D = P −1 AP =  .

. 
..
 ..
.
.
. 



0 0 · · · λm
Note that each column in the matrix
column
matrix
n
D
P
is equal to one of the eigenvectors of
is an eigenvector associated with the eigenvalue
contains the eigenvalues of
columns of
P
A
λn ,
A,
namely
and the resulting diagonal
on its diagonal. The order of the eigenvectors in the
is not important, as diering orders only changes the order of the eigenvalues
along the diagonal of
D.
It can also be shown that we can use the diagonalized matrix
a matrix.
P −1 AP = D
⇐⇒
A = P DP −1 .
D
to nd the
nth
power of
CHAPTER 4.
Letting
29
MATRIX FORMS AND PROPERTIES
1
D0 = D n
and
A0 = P (D0 )P −1 ,
(A0 )n = (P (D0 )P −1 )n
= (P (D0 )P −1 )(P (D0 )P −1 )(P (D0 )P −1 ) · · · (P (D0 )P −1 )
= P (D0 )(P −1 P )(D0 )(P −1 P ) · · · (P −1 P )(D0 )P −1
= P (D0 )n P −1 .
So
(A0 )n = A,
and again we will have a principal
nth
root for which we can write
1
An =
1
P D n P −1 .
4.3
Examples: Diagonalizing Matrix Forms of Complex
Numbers
We already know that we can use de Moivre's theorem to directly nd the
complex number
z ∈ C, or of a quaternion q ∈ H.
nth
roots of a
It would also be useful to see if the matrix
forms of the complex numbers and quaternions are also diagonalizable, and further if the
nth
roots of such diagonal matrices have the correct form.
4.3.1
Example 1:
2×2
Matrix of a Complex Number
The matrix form of a complex number contains only real numbers, so it makes sense to rst
try to diagonalize as a real matrix. Let
We want to nd a matrix
Let
X ∈ R2×2
C ∈ R2×2
such that

be the matrix form of a complex number.
X −1 CX

 a b
C=
.
−b a
is diagonal.
CHAPTER 4.
30
MATRIX FORMS AND PROPERTIES
Let


w x 
X=
.
y z
So,

X −1 =

1
 z −x

.
wz − xy −y w
Now multiplying them together yields




1
 z −x  a b  w x
X −1 CX =




wz − xy −y w
−b a
y z



zb − xa  w x
1
 za + xb
=



wz − xy −ya − wb −yb + wa
y z


(za + xb)x + (zb − xa)z 
1
 (za + xb)w + (zb − xa)y
=


wz − xy (−ya − wb)w + (−yb + wa)y (−ya − wb)x + (−yb + wa)z
which must be a diagonal matrix. This implies
wb)w + (−yb + wa)y = 0.
(za + xb)x + (zb − xa)z = 0
Simplifying these equations, we see
0 = (za + xb)x + (zb − xa)z
= xza + x2 b + z 2 b − xza
= x2 b + z 2 b
= (x2 + z 2 )b = 0
and
(−ya −
CHAPTER 4.
31
MATRIX FORMS AND PROPERTIES
and
0 = (−ya − wb)w + (−yb + wa)y
= −wya − w2 b − y 2 b + wya
= −w2 b − y 2 b
= (w2 + y 2 )(−b) = 0.
So, we nd that either
b = 0
w, x, y, z ∈ R, w, x, y, z = 0.
or both
x2 + z 2 = 0
and
w 2 + y 2 = 0,
meaning that when
Thus, the real matrix form of a non-real complex number is
not diagonalizable as a real matrix.
Since we note that the diagonalization failed in the real matrices, we should instead try
to diagonalize again in the complex matrices. Let
C ∈ C2×2
representation of a complex number.


 a b
C=
.
−b a
be a matrix in the form of a
CHAPTER 4.
MATRIX FORMS AND PROPERTIES
32
We verify that the matrix is diagonalizable by nding the eigenvalues.
a − λ
b
det(C − λI) = −b a − λ
= (a − λ)(a − λ) − (b)(−b)
= λ2 − 2λa + a2 + b2 = 0
=⇒ (λ − a)2 = −b2
√
λ−a =
−b2
= ±bi
=⇒
λ = a ± bi.
We have exactly two distinct eigenvalues when
b 6= 0,
which means that this form of
2×2
matrix is always diagonalizable. Also note that the eigenvalues of the matrix are the original
complex number and its conjugate. We continue with the eigenvectors.
For
λ = a + bi,
(A − λI)v = 0,

 
 
−bi b  x
0

  =  ,
−b −bi
y
0
CHAPTER 4.
33
MATRIX FORMS AND PROPERTIES
−bix + by = 0,
−bx − biy = 0.
Thus since we assume
Similarly, for
b 6= 0, y = ix
and the eigenspace of
a + bi
λ = a − bi,
(A − λI)v = 0,
 
 
0
 bi b  x

  =  ,
0
−b bi
y

bix + by = 0,
−bx + biy = 0.

Thus, the eigenspace of
a − bi
is spanned by

1
 .
−i
So, placing the eigenvectors as columns in a matrix, we have


1 1 
P =

i −i
and its inverse

P −1 =

1 1 i 

.
2 1 −i
is spanned by
 
1
 .
i
CHAPTER 4.
34
MATRIX FORMS AND PROPERTIES
So, we can use these to nd the diagonalized form of the matrix representation of any complex
number:






0  1 i  1
 a b  1 1  a + bi

=


 ,
−b a
i −i
0
a − bi
1 −i 2
where we note that the entries on the diagonal are the original complex number and its
conjugate. The roots of the resulting diagonal matrix can now be easily calculated by the
roots of the entries on the diagonal, and as noted earlier, these values will not be equal
unless the original number is equal to its complex conjugate.
These roots will be further
investigated in section 4.4.
4.3.2
Example 2:
For some quaternion
2×2
Matrix of a Quaternion
q ∈ H, q = a + bi + cj + dk ,
quaternion. We nd the eigenvalues of
let
Q ∈ C2×2
be a matrix form of a
Q:


 a + bi c + di
Q = 
,
−c + di a − bi


c + di 
(a + bi) − λ
det(Q − λI) = det 

−c + di
(a − bi) − λ
= (a − λ + bi)(a − λ − bi) − (c + di)(−c + di)
= a2 − 2aλ + λ2 + b2 + (ba − bλ − ba + bλ) i − −c2 − d2 + (−cd + cd) i
= λ2 − 2aλ + a2 + b2 + c2 + d2 .
CHAPTER 4.
35
MATRIX FORMS AND PROPERTIES
Thus, the characteristic equation is
(λ − a)2 + b2 + c2 + d2 = 0,
(λ − a)2 = −b2 − c2 − d2 ,
√
λ = a ± i b2 + c 2 + d 2 .
√
We can choose to represent the quantity
eigenvalues
a + ωi,
and
a − ωi,
b2 + c2 + d2
with
ω,
and we can diagonalize this matrix.
eigenvectors,
For
giving us the two distinct
λ = a + ωi,
(Q − λI)v = 0,
 
 
c + di  x
0
 bi − ωi

  =  ,
0
−c + di −bi − ωi
y

x (bi − ωi) + y(c + di) = 0,
x(−c + di) − y (−bi − ωi) = 0.
Now, to nd the
CHAPTER 4.
MATRIX FORMS AND PROPERTIES
So
(c + di)y = −(bi − ωi)x.
And the resulting eigenvector:


 c + di 
v = 
.
(−b + ω)i
Similarly, for
λ = a − ωi
 
 
c + di  x
0
 bi + ωi
  =  ,

0
−c + di −bi + ωi
y

x (bi + ωi) + y(c + di) = 0,
x(−c + di) − y (−bi + ωi) = 0.
So
(c + di)y = −(bi + ωi)x.
36
CHAPTER 4.
37
MATRIX FORMS AND PROPERTIES
And the resulting eigenvector:


 c + di 
v = 
.
(−b − ω)i
Now the eigenvectors yield the matrix


c + di 
 c + di
P =

(−b + ω)i (−b − ω)i
and its inverse


P −1 =
c + di 
1
 (b + ω)i
.

2ωi(c + di) (−b + ω)i −(c + di)
Note that for the inverse to exist, we must have
(c + di) 6= 0.
These are now used to nd the diagonalized form of the complex matrix representation
of any quaternion,
Qd ∈ C2×2


0 
a + ωi
P −1 QP = Qd = 

0
a − ωi
and we can now create
nth
roots as before.
CHAPTER 4.
4.3.3
Example 3:
We now look at the
Q ∈ C4×4
38
MATRIX FORMS AND PROPERTIES
4×4
4×4
Matrix of a Quaternion
matrix form of a quaternion, and see if it is diagonalizable. Let
be a matrix form of a quaternion. We nd the eigenvalues of

Q:

b
c
d
a


 −b a −d c 


Q = 
,
 −c d
a −b




−d −c b
a
a − λ
b
c
d −b a − λ −d
c
det(Q − λI) = −c
d
a − λ −b −d
−c
b
a − λ
= λ2 − 2aλ + a2 + b2 + c2 + d2 λ2 − 2aλ + a2 − b2 + c2 + d2
= 0.
CHAPTER 4.
39
MATRIX FORMS AND PROPERTIES
Thus,
(λ − a)2 + b2 + c2 + d2 = 0,
(λ − a)2 = −b2 − c2 − d2 ,
√
λ = a ± −b2 − c2 − d2 ,
or
(λ − a)2 − b2 + c2 + d2 = 0,
(λ − a)2 = b2 − c2 − d2 ,
√
λ = a ± b2 − c 2 − d 2 .
Here we see that when
b 6= 0,
we have four distinct eigenvalues,
a+
√
−b2 − c2 − d2 , a −
√
√
√
−b2 − c2 − d2 , a + b2 − c2 − d2 , and a − b2 − c2 − d2 , and when b = 0, the eigenvalues
√
√
a + −c2 − d2 , a − −c2 − d2 each have multiplicity 2, indicating that this matrix is also
diagonalizable.
We can use the representations
ω=
√
b2 + c2 + d2 , ϕ =
to write the eigenvalues in a more compact form:
For
−b2 + c2 + d2 , and γ = (c2 +d2 )
a + ωi, a − ωi, a + ϕi, a − ϕi.
λ = a − ωi,

√

−bd−cωi
− γ 


− −bc+dωi 


γ
v = 
.


1




0
CHAPTER 4.
For
40
MATRIX FORMS AND PROPERTIES
λ = a + ωi,


−bd+cωi
− γ 


− −bc−dωi 


γ
v = 
.


1




0
For
λ = a − ϕi,

c( −bd+cϕi
−
−γ
ϕi(bc−dϕi)
b(−γ)
)

d


 −bd+cϕi ϕi(bc−dϕi) 
−
+ b(−γ) 


−γ
v = 
.


ϕi
−b




1
For
λ = a + ϕi,

c( −bd−cϕi
+
−γ
ϕi(bc+dϕi)
b(−γ)
d

 −bd−cϕi
−
−

−γ
v = 

ϕi

b

1
We then construct the matrix
to generate a diagonal matrix.
P
and its inverse
)



ϕi(bc+dϕi) 
b(−γ) 
.



P −1
using the eigenvectors, and use them
CHAPTER 4.

P −1
− 2cϕi
− 2dωiϕi
γ
b(γ)
2dϕi
− 2cωiϕi
γ
b(γ)
4d2 ωiϕi
4c2 ωiϕi
−
−
(γ)2
(γ)2
 − 4c2 ωiϕi − 4d2 ωiϕi

(γ)2
(γ)2

2cϕi
2dωiϕi
−
− 2dϕi
− 2cωiϕi

γ
b(γ)
γ
b(γ)

2 ωiϕi
4d2 ωiϕi
4c2 ωiϕi
4d2 ωiϕi
−
−
−
 − 4c(γ)
2
(γ)2
(γ)2
(γ)2
=

− “ 2 2dωi 2 ” − “ 2 2cωi 2 ”
 (γ) − 4c ωiϕi − 4d ωiϕi
(γ) − 4c ωiϕi
− 4d ωiϕi

(γ)2
(γ)2
(γ)2
(γ)2

 “
2dωi
2cωi
“
”
”
2 ωiϕi
2
− 4d ωiϕi
(γ)2
(γ)2
(γ) − 4c
2 ωiϕi
2
− 4d ωiϕi
(γ)2
(γ)2
(γ) − 4c
4
0
0
2
2bc2 ωi
+ 2bd 2ωi
(γ)2
(γ)
2
2
− 4c ωiϕi
− 4d ωiϕi
(γ)2
(γ)2
2
2
− 2bc 2ωi − 2bd 2ωi
(γ)
(γ)
2
2
− 4c ωiϕi
− 4d ωiϕi
(γ)2
(γ)2
2 2
4
2c ϕi
4c d ϕi
2d ϕi
− b(−γ)(γ)
− b(−γ)(γ)
− b(−γ)(γ)
2
2 ωiϕi
− 4d ωiϕi
(γ)2
(γ)2
2c4 ϕi
4c2 d2 ϕi
2d4 ϕi
+ b(−γ)(γ)
+ b(−γ)(γ)
b(−γ)(γ)
2
2
− 4c ωiϕi
− 4d ωiϕi
(γ)2
(γ)2
2
2
− 2c ωiϕi
− 2d ωiϕi
(γ)2
(γ)2
2
2
− 4c ωiϕi
− 4d ωiϕi
(γ)2
(γ)2
2
2
− 2c ωiϕi
− 2d ωiϕi
(γ)2
(γ)2
2
2
− 4c ωiϕi
− 4d ωiϕi
(γ)2
(γ)2
− 4c







.






MATRIX FORMS AND PROPERTIES


ϕi(bc−dϕi)
ϕi(bc+dϕi)
c( −bd+cϕi
− b(−γ) )
c( −bd−cϕi
+ b(−γ) )
−γ
−γ
−bd−cωi
−bd+cωi
− γ
d
d
− γ



ϕi(bc−dϕi)
ϕi(bc+dϕi)
−bd−cϕi
− −bc+dωi − −bc−dωi − −bd+cϕi +

−
−

γ
γ
−γ
b(−γ)
−γ
b(−γ) 
P =
.


ϕi
ϕi
1
1
−b


b


0
0
1
1
41
CHAPTER 4.
42
MATRIX FORMS AND PROPERTIES
We write out the diagonal matrix
Qd ∈ C4×4


0
0
0 
a − ωi


 0

a
+
ωi
0
0


P −1 QP = Qd = 
,
 0
0
a − ϕi
0 




0
0
0
a + ϕi
and again we can nd roots of the diagonalized matrix.
4.4
Roots of Complex Numbers vs. Roots of Diagonal
Matrices
When de Moivre's formula is used to nd a root of a complex number
the root
1
zn
Considering
z,
the matrix form of
can easily be seen to be one root of the matrix form of the complex number
Z
as the matrix form of
the diagonalization of the matrix
z = eiθ
(using polar form for readability),
z.
Zd = P −1 ZP
1
1
Z , and z n
or
Zdn
as the principal
nth root of each, we have


iθ
0 
e
Zd = 
,
−iθ
0 e

1
in
θ
1
e
Zdn = 
0

0 
.
1
e−i n θ
We know that these are not the only possible roots of the matrix
each have
roots of
z
n distinct roots, any of the n2
and
z̄
on the diagonal of the
Zd .
In fact, since
z
and
z̄
matrices formed by all possible permutations of the
2 × 2 matrix form a valid matrix root of Zd .
Further,
it is not obvious that the roots of a matrix found from diagonalization are the only roots of
CHAPTER 4.
MATRIX FORMS AND PROPERTIES
the matrix. As an example, in the
2×2
real matrix space, we can construct an example of
a root of a diagonal matrix which is not itself a diagonal matrix.
2



1 x 
1 0

 =
.
0 −1
0 1
We can see that for any real matrix of this form



2
2
a b 
a ab + bd
=



,
2
0 d
0
d
when
a = −d,
the square of the matrix is


2
a

0
In general, we can see for any
2×2
43
0
.
d2
matrix whose square is diagonal,

 
2 

2
a b 
 a + bc ab + bd n 0 
=
,

 =
0 m
ac + cd d2 + bc
c d
CHAPTER 4.
44
MATRIX FORMS AND PROPERTIES
giving the set of equations
a2 + bc = n,
ab + bd = 0 ⇒ a = −d
or
b = 0,
ac + cd = 0 ⇒ a = −d
or
c = 0,
d2 + bc = m.
So,
b = 0, c 6= 0 ⇒ a = −d,
a2 = d2 = n = m,
c = 0, b 6= 0 ⇒ a = −d,
a2 = d2 = n = m,
b 6= 0, c 6= 0 ⇒ a = −d,
a2 + bc = d2 + bc = n = m,
b = c = 0 ⇒ a2 = n, d2 = m ∀a, d.
And conversely,
n = m ⇒ a2 + bc = d2 + bc
⇒ a2 = d 2
⇒ a = ±d,
CHAPTER 4.
45
MATRIX FORMS AND PROPERTIES
so,
a = d 6= 0 ⇒ b = c = 0
since
ab + bd = 0
and
ac + cd = 0,
a = −d 6= 0 ⇒ ab + bd = 0, ac + cd = 0, a2 + bc = d2 + bc = n = m ∀b, c,
a = d = 0 ⇒ ab + bd = 0, ac + cd = 0, bc = n = m ∀b, c.
Thus, a diagonal
2×2
matrix
A
has non-diagonal square roots if and only if it is a scalar
multiple of the identity matrix, ie.
4.5
A = nI .
Jordan Canonical Form
A matrix in Jordan canonical form, or Jordan normal form, is a block matrix of the form

J1


J2

J =




..
.
Jm








consisting of one or more Jordan blocks, where all entries outside of the blocks are zeros. Each
Jordan block is a matrix having zeroes everywhere except the diagonal and the superdiagonal,
CHAPTER 4.
MATRIX FORMS AND PROPERTIES
46
and the elements on the superdiagonal all consist of 1.


0 ··· 0
0
λm 1


0 λ
1
0
0
m




. 

..
.
.
0
0 λm
. 
.
Jm = 
 ..

..
 .

.
1
0




0
0
0
λm 1 




0
0
0 · · · 0 λm
Each
λm
represents an eigenvalue of the Jordan matrix, and each may or may not have
dierent values from other blocks. A
1×1
matrix can also be considered a Jordan block
even though it lacks a superdiagonal [48].
For any square matrix
A
there is a unique Jordan matrix decomposition
A = SJS −1 ,
where
S −1
is the matrix inverse of
S,
Given any complex square matrix
by nding a Jordan basis
bi,j
and
J
is a matrix in Jordan canonical form [49].
A, the matrix can be written in Jordan canonical form
for each Jordan block, where each Jordan basis satises [39, 40]
Abi,1 = λi bi,1
and
Abi,j = λi bi,j + bi,j−1 .
Much like diagonalized matrices, it is easy to relate a power of a matrix to a power of its
CHAPTER 4.
47
MATRIX FORMS AND PROPERTIES
Jordan canonical form,
A = SJS −1 ,
An =
SJS −1
n
SJS −1
SJS −1 · · · SJS −1
= SJ S −1 S J S −1 S · · · S −1 S JS −1
=
= SJ n S −1 ,
and a power of a Jordan matrix can also be easily calculated using the power of its Jordan
blocks.

n
J1


J2n

Jn = 




..
.
n
Jm




.



Also as with a diagonal matrix, any roots of a a Jordan matrix can be calculated using the
roots of its Jordan blocks. Given a Jordan Matrix
terms of the Jordan blocks from
J
(Ji0 )n = Ji .
where
(J 0 )n = J ,
we can write
J0
in
as:

0
J1


J20

J0 = 



where
J0

..
.
0
Jm




,



This would potentially allow us to nd an
nth
provided we can nd roots of its corresponding Jordan blocks.
root of any square matrix,
The next sections explore
CHAPTER 4.
48
MATRIX FORMS AND PROPERTIES
cases where the roots may not exist, and a criteria for knowing when the roots do exist.
4.6
Nilpotent Matrices
A nilpotent matrix is any matrix that when taken to some power
For an
m×m
matrix
n becomes the zero matrix.
A,
An = [0]
n
and we can call the minimum value of
nilpotent matrix
for which
An
is the zero matrix the index of the
A.
It can be shown that the eigenvalues of a nilpotent matrix are always 0. When nding
an eigenvalue
λ,
A~x = λ~x,
An~x = λn~x,
and since
An
is the zero matrix,
An~x = ~0
⇒
λn~x = ~0.
Because this will be true for any eigenvector
~x,
we must have
λ = 0.
The most obvious example of a nilpotent matrix is any strictly triangular matrix, with all
zeros along the diagonal. These are clearly not the only type of matrices that are nillpotent.
In fact it is not necessary that any entries in the matrix be zeroes, as we can construct
CHAPTER 4.
49
MATRIX FORMS AND PROPERTIES
examples such as


1
1
1
1
1
1
1



 



1
1
2
2
2
2

 



,
,


 1
1
1 

 3
−1 −1 
3
3
3


−2 −2 −2 
−6 −6 −6 −6




which are all nilpotent with index 2.
We see from the examples that it should be possible to construct a nilpotent
matrix of index 2 for any
m.
m×m
It would also be useful to show the maximum nilpotent index
that could exist for a given size
m.
Let
A
be a nilpotent
m×m
then not the zero matrix, so we can say there exists a vector
~x
matrix of index
such that
Ak−1~x
k . Ak−1
is
is non-zero.
It can then be seen that the k non-zero vectors
~x, A~x, A2~x, . . . , Ak−1~x
form a linearly independent set in
values
c0 , c1 , . . . , ck−1
Rm ,
which would imply
k ≤ m.
To wit, if we have scalar
which satisfy
c0~x + c1 A~x + · · · + ck−2 Ak−2~x + ck−1 Ak−1~x = ~0,
CHAPTER 4.
50
MATRIX FORMS AND PROPERTIES
then there will also be
k−1
equations
c0 A~x + c1 A2~x + · · · + ck−2 Ak−1~x + ck−1 Ak ~x = ~0,
c0 A2~x + c1 A3~x + · · · + ck−2 Ak ~x + ck−1 Ak+1~x = ~0,
.
.
.
c0 Ak−2~x + c1 Ak−1~x + · · · + ck−2 A2k−4~x + ck−1 A2k−3~x = ~0,
c0 Ak−1~x + c1 Ak ~x + · · · + ck−2 A2k−3~x + ck−1 A2k−2~x = ~0.
Since
Ak
and all higher powers of
c0 Ak−1~x = ~0,
implies
c1 = 0.
implying that
c0 = 0 .
A
are the zero matrix, the last equation above says
Then, the previous equation
c0 Ak−2~x + c1 Ak−1~x = ~0
Continuing in this manner through each equation, we nd that
c0 , c1 , . . . , ck−1
are all zero, which establishes linear independence of the vectors above [30]. Thus, we have
shown that for a nilpotent
m×m
matrix, the index
k ≤ m.
We can more easily see some properties of a nilpotent matrix by considering its Jordan
canonical form.
Since the Jordan canonical form of a nilpotent matrix always has zeros
on the main diagonal, and ones or zeros on the superdiagonal, the only non-trivial Jordan
canonical form of a
2×2
nilpotent matrix is


0 1
A=
.
0 0
CHAPTER 4.
For nilpotent
51
MATRIX FORMS AND PROPERTIES
3×3
2
matrices, we can have index 2 (B2
= 0)
3
or index 3 (B3
= 0).




0 1 0
0 0 0




 , B2 = 0 0 1 .
B3 = 
0
0
1








0 0 0
0 0 0
And for nilpotent
4
4 ( C4
4×4
2
matrices, we can have index 2 (C2
= 0),
3
index 3 (C3


0 0 0
0


0
0 1 0


 , C2 = 
0
0 0 1




0
0 0 0

0 0 0

0 0 0

.
0 0 1


0 0 0
= 0),
or index
= 0).

0

0

C4 = 
0


0


1 0 0
0


0
0 1 0


 , C3 = 
0
0 0 1




0
0 0 0
By this representation, it seems clear that the index of the nilpotent is related to the number
of Jordan blocks, and hence to the dimension of the eigenspace of
the
4×4
(x
~1
= [1, 0, 0, 0]), C3
examples above, we note that
has (x
~1
C4
0, ie.
the null space. From
has a nullspace of dimension
= [1, 0, 0, 0], x~2 = [0, 1, 0, 0])),
and
C2
1,
has (x
~1
generated by
= [1, 0, 0, 0],
x~2 = [0, 1, 0, 0]), x~3 = [0, 0, 1, 0])).
We would also like to take a quick look at whether we can calculate roots of a nilpotent
matrix. We start with the Jordan Canonical form of a


0 1 
A=
.
0 0
2×2
nilpotent matrix
CHAPTER 4.
MATRIX FORMS AND PROPERTIES
We want to nd some matrix
A0
where
52
A02 = A

2 
 

2
a b 
 a + bc ab + bd 0 1
A02 = 
 =
=
.
c d
ca + dc cb + d2
0 0
So, we have the series of equations
a2 + bc = 0,
ab + bd = 1
⇒ abc = c − bcd,
ca + dc = 0
⇒ abc = −bcd,
cb + d2 = 0.
Thus,
c = 0,
a2 = 0,
a = 0,
d2 = 0,
d = 0,
Contradiction, since ab + bd = 0.
Thus, we see there is no solution to this system of four equations with four unknowns, and
thus there are no square roots of a
2×2
nilpotent matrix.
CHAPTER 4.
53
MATRIX FORMS AND PROPERTIES
We next look at the Jordan Canonical forms of a


0 1 0



A=
0 0 1


0 0 0
where we want to nd some matrix
A0
3×3
nilpotent matrix


0 0 0



or A = 
0 0 1 ,


0 0 0
where
A02 = A or A03 = A.
Utilizing the Mathematica
software to solve the resulting systems of nine equations with nine unknowns also results in
no solution, and thus there are no square roots or cube roots of a
Looking further at the
4×4
1 0
0 1
0 0
0 0

0

0

,
1


0
AAAA = (AA)(AA) = [0], we can consider the matrix B = AA, where B 2 = [0].
the matrix
B.
nilpotent matrix.
nilpotent matrices, we notice that for the matrix

0

0

A=
0


0
since
3×3
B
is nilpotent, and the matrix
A
Thus,
is considered to be a square root of the matrix
This can be generalized based on the associative property of matrix multiplication for
larger matrices as well.
Theorem 4.6.1.
Let A be a nilpotent m × m matrix of index m, and let p, n ∈ Z, where
p, n > 1, such that pn = m. Then, there exists some nilpotent m × m matrix A0 of index p
where An = A0 . Thus, we can say that A is an nth root of A0 .
CHAPTER 4.
4.7
54
MATRIX FORMS AND PROPERTIES
Existence of Roots
We have already noted that there seems to be a relation between the dimension of the
nullspace of a nilpotent matrix and its index.
In fact, relations between dimensions of
nullspaces can be used to determine the existence of
of a matrix
A,
nth roots.
We utilize the ascent sequence
which is the sequence
di = dim Null Ai − dim Null Ai−1
; i = 1, 2, . . . .
Psarrakos [36] shows the following:
Theorem 4.7.1 (Psarrakos).
Given a complex m × m matrix A ∈ Cm×m , A has
an nth root if and only if for every integer ν ≥ 0, the ascent sequence of A has
no more than one element between nν and n(ν + 1).
and a corollary,
Corollary 4.7.2 (Psarrakos).
Let d1 , d2 , d3 , . . . be the ascent sequence of a sin-
gular complex matrix A.
(i) If d2 = 0, then for every integer n > 1, A has an nth root.
(ii) If d2 > 0, then for every integer n > d1 , A has no nth roots.
Now that we see a method for knowing if
nth
roots exist, we look for methods which
might provide us with multiple roots, should they exist.
Chapter 5
Hypercomplex Numbers
Knowing that it is possible to nd multiple roots in the complex numbers and in the quaternions, it makes sense to next look for other number systems for which multiple roots can be
found where we might be able to nd a ring isomorphism with the matrices. The area of
Hypercomplex Numbers oers many number systems in which we can begin looking.
A basic denition for a hypercomplex number is a number having properties which depart
from those of the real or complex numbers [45].
More precisely, a hypercomplex number
system in a nite dimensional algebra over the reals which is unital and distributive, but not
necessarily associative, with elements generated by real number coecients
for some basis
1, i0 , i1 , . . . , in ,
where each basis is conventionally normalized such that
−1, 0, 1
[28].
5.1
Multicomplex Numbers
The multicomplex numbers are a sequence of number systems
C0
is taken to be the real number system, and
for each
n > 0, in
(a0 , a1 , . . . , an )
is some square root of
−1.
55
Cn
i2k ∈
dened inductively, where
Cn+1 = {z = x + yin+1 : x, y ∈ Cn },
where
A further requirement is that multiplication
CHAPTER 5.
56
HYPERCOMPLEX NUMBERS
on the imaginary units must be commutative, such that for any
this denition,
Cn
C1
is the complex number system,
is a multicomplex number system of order
5.1.1
n
C2
n 6= m, in im = im in .
Under
is the bicomplex number system, and
[35].
Bicomplex Numbers
The bicomplex numbers were developed by Corrado Segre [42], using two commuting square
roots of
−1,
h2 = i2 = −1,
whose product would then have the square
(hi)2 = (ih)2 = 1.
The algebra also contains idempotent values
g=
5.1.2
1 + hi
2
g∗ =
1 − hi
.
2
Tessarines
The tessarines were rst described by James Cockle [9, 10, 11, 12, 13, 14] as an algebra
similar to the quaternions, but which have the form
t = w + xα + yβ + zγ,
where
w, x, y, z ∈ R
and
α2 = −1,
β 2 = +1,
αβ = βα = γ,
γ 2 = −1
CHAPTER 5.
A tessarine
t = w + xi + yj + zk
i = α, j = β, k = γ , α, β, γ
numbers,
57
HYPERCOMPLEX NUMBERS
(using more modern notation for complex numbers:
as listed above) can also be written in terms of two complex
t = (w + xi) + (y + zi)j ,
since
ij = k ,
so we can write
t = a + bj , a, b ∈ C.
We can use this representation to show a mapping into the
f : T 7→ C2×2
2×2
complex matrices


a b 
t = a + bj 7→ 

b a
and


 c d
s = c + dj →
7 
,
d c
so we see

 

 
a + c b + d a b   c d
t + s = (a + c) + (b + d)j 7→ 

+
=
d c
b a
b+d a+c
and

 


ac + bd bc + ad a b   c d
ts = (ac + bd) + (ad + bc)j 7→ 
=

.
bc + ad ac + bd
b a
d c
We can easily see that for any complex
2×2

matrix of the appropriate form,

a b 

,
b a
there will exist some
all
a, b ∈ C,
t ∈ T , t = a + bj .
So, the inverse mapping
f 0 : C2×2 7→ T
and we can see that this mapping is an isomorphism.
is dened for
CHAPTER 5.
5.2
58
HYPERCOMPLEX NUMBERS
Split-Complex Numbers
We can examine the non-real root of
+1 seen above in the Tessarines and Bicomplex Numbers
apart from the usual complex unit by examining the quantity
z = a + bj , j 2 = +1,
rst
examined by Cockle [14], they are often referred to currently as a split complex numbers
[38] or Hyperbolic numbers [43]. Much as seen earlier in the bicomplex numbers, the split
complex numbers will have idempotent values
z=
1+j
,
2
z∗ =
1−j
.
2
The split complex numbers also contain zero divisors.
product of any two split complex numbers
We can nd them by setting the
z1 = a1 + b1 j , z2 = a2 + b2 j
z1 z2 = (a1 + b1 j)(a2 + b2 j)
= a1 a2 + a1 b 2 j + a2 b 1 j + b 1 b 2
= (a1 a2 + b1 b2 ) + (a1 b2 + a2 b1 )j
= 0.
Setting the real and imaginary parts both to zero, we have:
a1 a2 + b1 b2 = 0,
a1 b2 + a2 b1 = 0.
to zero:
CHAPTER 5.
59
HYPERCOMPLEX NUMBERS
So
a1 =
−b1 b2
a2
and a1 =
−a2 b1
b2
if
a2 , b2 6= 0.
Thus
−b1 b2
−a2 b1
=
,
a2
b2
−b1 b22 = −a22 b1 ,
b22 = a22 .
So we see, the split complex number
z = a + bj
where
a = ±b
is a zero divisor for any other
split complex number.
In the
2 × 2 matrix space, there are a few dierent matrices that can be used to represent
a non-real root of
+1.
Looking at diagonal matrices, we can see that a matrix that has 1
and -1 on the diagonal will have a square that contains all 1's on the diagonal, and so is the
image of the real +1.


2 
2 
1 0
1 0 
−1 0
.

 =
 =
0 1
0 −1
0 1
Another matrix which squares to a diagonal matrix of ones is an anti-diagonal matrix of
ones:

2 

0 1
1 0

 =
.
1 0
0 1
Since this representation is orthogonal to the representation of a real number, we can use
CHAPTER 5.
60
HYPERCOMPLEX NUMBERS
this in mapping from the split complex numbers into the real
2×2
matrix space. If
a, b ∈ R


a b 
z = a + bj 7→ 
.
b a
5.3
Dual Numbers
As with the complex numbers (z
a + bj, j 2 = +1),
In the
2×2
= a + bi, i2 = −1)
it is possible to dene a number
and the split-complex numbers (z
z = a + b
where
is nilpotent,
matrix space, there are several matrices which could represent
2 = 0
=
[6].
, namely any
nilpotent matrix. The easiest example is any strictly triangular matrix, where all values on
the diagonal are 0.

2 
2 

0 1
0 0
0 0

 =
 =
.
0 0
1 0
0 0
Then the mapping


a b 
z = a + b 7→ 

0 a
is an isomorphism.
5.4
Split-Quaternions
The split-quaternions, which were rst described as the coquaternions by James Cockle [10],
are numbers of the form
q = w + xi + yj + zk
where
ij = k = −ji, jk = −i = −kj, ki = j = −ik, ijk = 1,
i2 = −1, j 2 = +1, k 2 = +1.
CHAPTER 5.
A split quaternion has conjugate
w 2 + x2 − y 2 − z 2 .
q̄ = w − xi − yj − zk ,
A split quaternion
q
q ∗ q −1 = q −1 ∗ q = 1,
and multiplicative modulus
q̄
,
q q̄
which exists when the multiplicative modulus
We can also write a split quaternion in terms of two complex numbers
b = y + zi,
so
q = a + bj .
q q̄ =
will have a multiplicative inverse
q −1 =
such that
61
HYPERCOMPLEX NUMBERS
q q̄ 6= 0.
a = w + xi,
When written this way, we can dene a mapping to the
2×2
complex matrix space


a b 
q = a + bj 7→ 
,
b̄ ā
where
a, b ∈ C,
and
ā, b̄
are the complex conjugates of
a
and
b.
Written out explicitly, this
gives


w + xi y + zi 
q = (w + xi) + (y + zi)j 7→ 
.
y − zi w − xi
Recall now, however, the real
2×2 matrix representations of each non-real component, where
we have one real matrix which can represent a root of
non-real root of
+1.
−1,
and two that can can represent a
We then consider the following mappings


 0 1
i 7→ 
,
−1 0


0 1
j 7→ 
,
1 0
CHAPTER 5.
62
HYPERCOMPLEX NUMBERS


1 0 
k 7→ 
,
0 −1
and see that


 

0 1 1 0  0 −1


=
,
1 0
0 −1
1 0
noting that this corresponds to
jk = −i
as seen in the coquaternions. We further note that
these matrices together with the identity matrix form a basis for the
Then for any arbitrary real
2×2
matrix, with
a, b, c, d ∈ R,
2×2
real matrices.
we have the mapping


(b + c)
(a − d)
(a + d) (b − c)
a b 
+
i+
j+
k
7 q=

→
2
2
2
2
c d
which forms a ring isomorphism [29].
5.4.1
Roots of Split-Quaternions
The split quaternions can also be represented in a polar form [33] which can be used to
exhibit a version of the de Moivre formula and subsequently used to nd roots[32]. The de
Moivre formula for split quaternions is dened piecewise depending on the properties of a
split quaternion
N (q) =
q = w + xi + yj + zk .
p
|w2 + x2 − y 2 − z 2 |.
the split quaternion
q
is called
The split quaternion
q
will have norm
Considering the multiplicative modulus
spacelike
if
q q̄ < 0, timelike
if
q q̄ > 0,
N (q) dened as
q q̄ = w2 +x2 −y 2 −z 2 ,
and
lightlike
if
q q̄ = 0,
with these labels stemming from how these quantities are used in physics. Note that under
these labels, spacelike and timelike split quaternions will have multiplicative inverses, but
lightlike quaternions will have no inverses.
Further characterization of a split quaternion is taken by splitting it into its scalar part
Sq = w,
and its vector part
V~q = xi + yj + zk ,
where the vector part is identied with the
CHAPTER 5.
63
HYPERCOMPLEX NUMBERS
Minkowski 3-space. The Lorentzian inner product of the vector part with itself,
−x2 + y 2 + z 2 ,
is called spacelike if
hV~q , V~q iL > 0,
timelike if
hV~q , V~q iL < 0,
hV~q , V~q iL =
or lightlike if
hV~q , V~q iL = 0.
It is then possible to write spacelike or timelike split quaternions in polar form, and
extend de Moivre's formula for the split-quaternions as follows:
We can write a timelike split quaternion with a spacelike vector part in the form
q = N (q) (cosh θ + ~ε sinh θ) ,
where
|w|
cosh θ =
, sinh θ =
N (q)
and
~ε 2 = ~ε ∗ ~ε = 1.
p
−x2 + y 2 + z 2
xi + yj + zk
, ~ε = p
N (q)
−x2 + y 2 + z 2
De Moivre's formula is then written as
q n = (N (q))n (cosh nθ + ~ε sinh nθ) .
The equation
wn = q
will have only one root
p
θ
θ
n
,
w = N (q) cosh + ~ε sinh
n
n
because the hyperbolic cosine and the hyperbolic sine are not periodic.
For a timelike split quaternion with a timelike vector part, we write
q = N (q) (cos θ + ~ε sin θ) ,
where
w
cos θ =
, sin θ =
N (q)
p
x2 − y 2 − z 2
xi + yj + zk
, ~ε = p
N (q)
x2 − y 2 − z 2
CHAPTER 5.
and
64
HYPERCOMPLEX NUMBERS
~ε 2 = ~ε ∗ ~ε = −1.
De Moivre's formula is then written as
q n = (N (q))n (cos nθ + ~ε sin nθ) .
The equation
wn = q
will have
n
roots
p
θ + 2mπ
θ + 2mπ
n
wm = N (q) cos
+ ~ε sin
,
n
n
where
m = 0, 1, 2, . . . , n − 1.
For a timelike split quaternion with a lightlike vector part, we write
q = 1 + ~ε,
where
~ε
is a null vector. We can still write a form of de Moivre's formula as
q n = 1 + n~ε.
There will be innitely many roots of the equation
wn = q .
For a spacelike split quaternion, we write
q = N (q) (sinh θ + ~ε cosh θ) ,
where
w
, cosh θ =
sinh θ =
N (q)
and
~ε
p
−x2 + y 2 + z 2
xi + yj + zk
, ~ε = p
N (q)
−x2 + y 2 + z 2
is a spacelike vector. De Moivre's formula is then written as
q n = (N (q))n (sinh nθ + ~ε cosh nθ) ,
n is odd;
CHAPTER 5.
65
HYPERCOMPLEX NUMBERS
q n = (N (q))n (cosh nθ + ~ε sinh nθ) ,
The equation
wn = q
will have no roots if
n
n is even.
is an even number, and if
n
is odd it will have
one root
p
θ
θ
n
w = N (q) sinh + ~ε cosh
.
n
n
So, for a split quaternion
q = w + xi + yj + zk ,
w 2 + x2 − y 2 − z 2 < 0
multiple roots, when
w 2 + x2 − y 2 − z 2 > 0
when
we can nd a single
nth
root if
n
we can nd
is odd, and
otherwise we will not be able to nd roots. Re-stated in terms of the mapping to a
real matrix,
2×2
a, b, c, d ∈ R


(b + c)
(a − d)
(a + d) (b − c)
a b 
+
i+
j+
k,
7 q=

→
2
2
2
2
c d
the quantity
w 2 + x2 − y 2 − z 2
=
a+d
2
maps to
2
+
root if
5.5
n
2
−
b+c
2
2
−
a−d
2
2
(a2 + 2ad + d2 ) + (b2 − 2bc + c2 ) − (b2 + 2bc + c2 ) − (a2 − 2ad + d2 )
4
=
So, when
b−c
2
ad − bc > 0
4ad − 4bc
= ad − bc.
4
we can nd multiple roots, when
ad − bc < 0
we can nd a single
nth
is odd, and otherwise we will not be able to nd roots using this method.
Biquaternions
The biquaternions, or complexied quaternions, are numbers of the form
where the elements
i, j, k
multiply as in the quaternions, and
q = w+xi+yj +zk ,
w, x, y, z ∈ C
are normal
CHAPTER 5.
66
HYPERCOMPLEX NUMBERS
complex numbers [15, 22, 23]. The normal complex unit is denoted
h,
and we have
hi = ih, hj = jh, hk = kh
since
h
is a scalar with respect to the quaternion elements in this context.
A biquaternion has two forms of conjugate [41]. First is the
complex conjugate,
i.e. the
complex conjugate of the complex components of the quaternion,
q ∗ = w∗ + x∗ i + y ∗ j + z ∗ k,
and for two biquaternions,
(pq)∗ = p∗ q ∗ .
Next is the
quaternion conjugate
q̄ = w − xi − yj − zk,
where the conjugate rule for quaternions
q̄ p̄ = pq
and the generalization to more than two quaternions also applies, and a biquaternion commutes with its quaternion conjugate
q q̄ = q̄q.
The quaternion conjugate is used in nding the inverse of a biquaternion
q −1 =
q̄
q q̄
CHAPTER 5.
67
HYPERCOMPLEX NUMBERS
which exists when the modulus
q q̄ 6= 0,
and is also seen when nding a polar form of a
biquaternion which is used in the next section for nding roots.
We can create a mapping from the biquaternions to the complex
2×2
matrices by rst
mapping the elements


h 0 
i 7→ 
,
0 −h


 0 1
j 7→ 
,
−1 0


 0 h
k→
7 
,
h 0
and see that



 
h 0   0 1  0 h
,


=
h 0
0 −h
−1 0
which corresponds to
ij = k
in quaternion multiplication. We can then dene the mapping
q = w + xi + yj + zk
where for any complex
2×2
matrix
7→

 

a b   w + xh y + zh 
A=
,
=
−y + zh w − xh
c d
A ∈ C2×2 ,
there are values for
(a + d)
,
2
−h(a − d)
x =
,
2
(b − c)
y =
,
2
−h(b + c)
z =
,
2
w =
w, x, y, z ∈ C,
CHAPTER 5.
68
HYPERCOMPLEX NUMBERS
such that the matrix ring is isomorphic to the biquaternion ring [17].
5.5.1
Roots of Biquaternions
There are two possible polar forms of a biquaternion, as the biquaternions have two possibilities for roots of -1: the complex root
h,
or a biquaternion root of -1. A biquaternion root
of -1 is found by taking any pure biquaternion, ie. a biquaternion with its scalar part
with modulus
√
q q̄ 6= 0,
w = 0,
and dividing it by its modulus.
q
√
q q̄
2
q2
q q̄
q
=
q̄
0 + xi + yj + zk
=
0 − xi − yj − zk
=
= −1.
The rst polar form of a biquaternion is known as the
Complex polar form,
written as
q = Q exp(hΨ)
= Q(cos Ψ + h sin Ψ),
where
Q and Ψ are both quaternions.
If we write the original biquaternion
q = w+xi+yj+zk
in its expanded form
q = (wr + wi h) + (xr + xi h)i + (yr + yi h)j + (zr + zi h)k,
where
wr , wi , xr , xi , yr , yi , zr , zi ∈ R,
we see that the biquaternion can be rewritten in the
CHAPTER 5.
69
HYPERCOMPLEX NUMBERS
form of two quaternions
q = (wr + xr i + yr j + zr k) + (wi + xi i + yi j + zi k)h
= qr + qi h,
where
qr , qi ∈ H.
We can then see the complex modulus
cos Ψ =
qr
,
Q
sin Ψ =
Q=
p
qr2 + qi2 ,
and
qi
.
Q
We note that for this form, the modulus and exponential do not commute, meaning it is not
evident that a form of de Moivre's formula is possible.
The second form is known as the
Hamilton polar form,
written as
q = R exp(ξΘ)
= R(cos Θ + ξ sin Θ),
where
R =
p
w 2 + x2 + y 2 + z 2 ,
w
,
R
p
x2 + y 2 + z 2
sin Θ =
,
R
xi + yj + zk
ξ = p
,
x2 + y 2 + z 2
cos Θ =
CHAPTER 5.
70
HYPERCOMPLEX NUMBERS
and we see that
ξ
is a biquaternion root of
−1.
and also note that the form exists only when
form of the complex modulus
R
We note that both
R 6= 0.
R
and
Θ
are complex,
We can also utilize the exponential
to write the biquaternion as
q = r exp(hφ) exp(ξΘ),
where
R = r exp(hφ),
and the exponentials commute since
h
commutes with
We can now take this form with two commuting exponentials to an
nth
ξ
[41].
power,
q n = (r exp(hφ) exp(ξΘ))n
= (r exp(hφ) exp(ξΘ))(r exp(hφ) exp(ξΘ)) . . . (r exp(hφ) exp(ξΘ))
{z
}
|
n
= (r)n (exp(hφ))n (exp(ξΘ)n ,
since
h
and
Θ
commute with
ξ
= rn exp(hnφ) exp(ξnΘ)
= rn (cos(nφ) + h sin(nφ))(cos(nΘ) + ξ sin(nΘ)),
giving us an implementation of de Moivre's formula that can be used for biquaternions.
Similarly, for nding a root of a biquaternion, we will have
q
where
k1 , k2
0
φ + 2πk1
Θ + 2πhk2
= r exp h
exp ξ
n
n
1
φ + 2πk1
φ + 2πk1
= r n cos
+ h sin
n
n
Θ + 2πhk2
Θ + 2πhk2
cos
+ ξ sin
n
n
1
n
are integers, and
(q 0 )n = q .
We can see that these are likely not the only possible
CHAPTER 5.
71
HYPERCOMPLEX NUMBERS
roots of the biquaternion
q,
since the Hamiltonian polar form is not the only possible polar
form of the biquaternion.
So, we know that we can nd roots of a biquaternion
modulus
p
R = w2 + x2 + y 2 + z 2 6= 0,
q = w + xi + yj + zk
and in terms of a complex
2×2
matrix


(b − c)
−h(b + c)
(a + d) −h(a − d)
a b 
A=
+
i+
j+
k,
7 q=
→
2
2
2
2
c d
the modulus becomes
s
r
=
and we can
a+d
2
2
+
−h(a − d)
2
2
+
b−c
2
2
+
−h(b + c)
2
2
(a2 + 2ad + d2 ) − (a2 − 2ad + d2 ) + (b2 − 2bc + c2 ) − (b2 + 2bc + c2 )
4
r
4ad − 4bc √
=
= ad − bc,
4
√
nd roots if
ad − bc 6= 0.
when its
Chapter 6
Conclusion
There are multiple iterative methods of nding some
nth
root of a real or complex
m×m
matrix, though such methods may fail to nd a root even if such a root exists, and will
only nd a single root. If an
have
m×m
nth roots, for any integer n.
For non-diagonalizable matrices, as seen specically in the
nilpotent matrices, there may be no
n.
matrix is diagonalizable, it has been shown to always
nth
roots, or roots may exist only for specic values of
Utilizing the isomorphism between the
2×2
real matrices and the split-quaternions, we
have a form of de Moivre's formula which can be used to nd roots of a
Similarly we have an isomorphism from the
2×2
2×2 complex matrices into the biquaternions, for
which we can use de Moivre's formula to nd some of the possible roots of a
matrix.
real matrix.
2×2
complex
The octonions have a form of de Moivre's formula [7], but have no isomorphism
with a matrix space as they are not associative. We can use these results to point towards
a potential method of nding roots for larger matrix spaces, where we attempt to nd some
isomorphism into a hypercomplex space for which a form of de Moivre's formula can be
found. One possible area for further research into this idea is the Cliord algebras, which
are a set of hypercomplex algebras which maintain associativity at all levels of complexity
[34].
72
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