Download Appendix E An Introduction to Matrix Algebra

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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
APPENDIX
An Introduction
to Matrix Algebra
E
E.1 DEFINITIONS
In many situations, we must deal with rectangular arrays of numbers or functions.
The rectangular array of numbers (or functions)
a11
a
A = D 21
o
am1
a12
a22
o
am2
p
p
p
a1n
a2n
T
o
amn
(E.1)
is known as a matrix. The numbers aij are called elements of the matrix, with the subscript i denoting the row and the subscript j denoting the column.
A matrix with m rows and n columns is said to be a matrix of order (m, n) or alternatively called an m n (m-by-n) matrix.When the number of the columns equals
the number of rows (m n), the matrix is called a square matrix of order n. It is common to use boldfaced capital letters to denote an m n matrix.
A matrix comprising only one column, that is, an m 1 matrix, is known as a
column matrix or, more commonly, a column vector. We will represent a column vector with boldfaced lowercase letters as
y1
y
y = D 2T
o
ym
(E.2)
Analogously, a row vector is an ordered collection of numbers written in a row—
that is, a 1 n matrix. We will use boldfaced lowercase letters to represent vectors.
Therefore a row vector will be written as
z = 3z1
z2
p
zn 4,
(E.3)
with n elements.
A few matrices with distinctive characteristics are given special names. A square
matrix in which all the elements are zero except those on the principal diagonal, a11,
a22, . . . , ann, is called a diagonal matrix. Then, for example, a 3 3 diagonal matrix
would be
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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Appendix E
An Introduction to Matrix Algebra
0
0
b22
0 S.
0 b33
b11
B= C 0
0
(E.4)
If all the elements of a diagonal matrix have the value 1, then the matrix is known as
the identity matrix I, which is written as
1
0
I=D
o
0
0
1
o
0
p
p
p
p
0
0
T.
o
1
(E.5)
When all the elements of a matrix are equal to zero, the matrix is called the zero, or
null matrix. When the elements of a matrix have a special relationship so that aij aji, it is called a symmetrical matrix. Thus, for example, the matrix
3
H = C -2
1
1
4S
8
-2
6
4
(E.6)
is a symmetrical matrix of order (3, 3).
E.2 ADDITION AND SUBTRACTION OF MATRICES
The addition of two matrices is possible only for matrices of the same order. The sum
of two matrices is obtained by adding the corresponding elements.Thus if the elements
of A are aij and the elements of B are bij, and if
C A B,
(E.7)
then the elements of C that are cij are obtained as
cij aij bij.
(E.8)
For example, the matrix addition for two 3 3 matrices is as follows:
2
C = C1
0
1
-1
6
0
8
3S + C1
2
4
2
3
2
1
10
0S = C 2
1
4
3
2
8
1
3 S.
3
(E.9)
From the operation used for performing the operation of addition, we note that the
process is commutative; that is,
A B B A.
(E.10)
Also we note that the addition operation is associative, so that
(A B) C A (B C).
(E.11)
To perform the operation of subtraction, we note that if a matrix A is multiplied
by a constant a, then every element of the matrix is multiplied by this constant.Therefore we can write
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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Section E.3
Multiplication of Matrices
aa11
aa12
aA = D
o
aam1
aa12
aa22
o
aam2
p
p
p
aa1n
aa2n
T.
o
aamn
7
(E.12)
Then to carry out a subtraction operation, we use a 1, and A is obtained by multiplying each element of A by 1. For example,
2
4
C=B-A=B
1
6
R-B
2
3
1
-4
R=B
1
1
0
R.
1
(E.13)
E.3 MULTIPLICATION OF MATRICES
The multiplication of two matrices AB requires that the number of columns of A
be equal to the number of rows of B. Thus if A is of order m n and B is of order n
q, then the product is of order m q. The elements of a product
C AB
(E.14)
are found by multiplying the ith row of A and the jth column of B and summing these
products to give the element cij. That is,
q
cij = ai1b1j + ai2b2j + p + aiqbqj = a aikbkj.
(E.15)
k=1
Thus we obtain c11, the first element of C, by multiplying the first row of A by the first
column of B and summing the products of the elements. We should note that, in general, matrix multiplication is not commutative; that is
AB BA.
(E.16)
Also we note that the multiplication of a matrix of m n by a column vector (order
n 1) results in a column vector of order m 1.
A specific example of multiplication of a column vector by a matrix is
a
x = Ay = B 11
a21
a12
a22
y1
a13
1a y + a12y2 + a13y3 2
R C y2 S = B 11 1
R.
a23
1a21y1 + a22y2 + a23y3 2
y3
(E.17)
Note that A is of order 2 3, and y is of order 3 1. Therefore the resulting matrix
x is of order 2 1, which is a column vector with two rows. There are two elements
of x, and
x1 (a11y1 a12y2 a13y3)
(E.18)
is the first element obtained by multiplying the first row of A by the first (and only)
column of y.
Another example, which the reader should verify, is
2
-1
C = AB = B
3
-1
RB
2 -1
2
7
R=B
-2
-5
6
R.
-6
(E.19)
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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Appendix E
An Introduction to Matrix Algebra
For example, the element c22 is obtained as c22 1(2) 2(2) 6.
Now we are able to use this definition of multiplication in representing a set of
simultaneous linear algebraic equations by a matrix equation. Consider the following set of algebraic equations:
3x1 2x2 x3 u1,
2x1 x2 6x3 u2,
4x1 x2 2x3 u3.
(E.20)
We can identify two column vectors as
x1
x = C x2 S
x3
u1
u = C u2 S.
u3
and
(E.21)
Then we can write the matrix equation
Ax u,
(E.22)
where
3
A = C2
4
2
1
-1
1
6 S.
2
We immediately note the utility of the matrix equation as a compact form of a set of
simultaneous equations.
The multiplication of a row vector and a column vector can be written as
xy = 3x1 x2
p
y1
y2
xn 4 D T = x1 y1 + x2 y2 + p + xn yn.
o
yn
(E.23)
Thus we note that the multiplication of a row vector and a column vector results in
a number that is a sum of a product of specific elements of each vector.
As a final item in this section, we note that the multiplication of any matrix by
the identity matrix results in the original matrix, that is, AI A.
E.4 OTHER USEFUL MATRIX OPERATIONS AND DEFINITIONS
The transpose of a matrix A is denoted in this text as AT. One will often find the notation A' for AT in the literature. The transpose of a matrix A is obtained by interchanging the rows and columns of A. For example, if
6
A= C 1
-2
0
4
3
2
1 S,
-1
then
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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Section E.4
9
Other Useful Matrix Operations and Definitions
6
AT = C 0
2
1
4
1
-2
3 S.
-1
(E.24)
Therefore we are able to denote a row vector as the transpose of a column vector and
write
xT = 3x1
x2
p
xn 4.
(E.25)
Because xT is a row vector, we obtain a matrix multiplication of xT by x as follows:
xTx = 3x1
x2
p
x1
x2
xn 4 D T = x21 + x22 + p + x2n.
o
xn
(E.26)
Thus the multiplication xTx results in the sum of the squares of each element of x.
The transpose of the product of two matrices is the product in reverse order of
their transposes, so that
(AB)T BTAT.
(E.27)
The sum of the main diagonal elements of a square matrix A is called the trace
of A, written as
tr A a11 a22 … ann.
(E.28)
The determinant of a square matrix is obtained by enclosing the elements of the
matrix A within vertical bars; for example,
det A = 2
a11 a12
2 = a11a22 - a12a21.
a21 a21
(E.29)
If the determinant of A is equal to zero, then the determinant is said to be singular.
The value of a determinant is determined by obtaining the minors and cofactors of
the determinants. The minor of an element aij of a determinant of order n is a determinant of order (n 1) obtained by removing the row i and the column j of the original determinant.The cofactor of a given element of a determinant is the minor of the
element with either a plus or minus sign attached; hence
cofactor of aij aij (1)ijMij,
where Mij is the minor of aij. For example, the cofactor of the element a23 of
a11
det A = 3 a21
a31
a12
a22
a32
a13
a23 3
a33
(E.30)
is
a23 = 1-12 5M23 = - 2
a11
a31
a12
2.
a32
(E.31)
The value of a determinant of second order (2 2) is
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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Appendix E
An Introduction to Matrix Algebra
a11
a21
2
a12
2 = 1a11a22 - a21a12 2.
a22
(E.32)
The general nth-order determinant has a value given by
n
det A = a aijaij
with i chosen for one row,
(E.33)
with j chosen for one column.
(E.33)
j=1
or
n
det A = a aijaij
i=1
That is, the elements aij are chosen for a specific row (or column), and that entire row
(or column) is expanded according to Eq. (E.33). For example, the value of a specific
3 3 determinant is
2
det A = det C 1
2
=2 2
0
1
3
0
1
5
1S
0
1
3
2 -1 2
0
1
5
3
2 +2 2
0
0
= 21-12 - 1-52 + 2132 = 9,
5
2
1
(E.34)
where we have expanded in the first column.
The adjoint matrix of a square matrix A is formed by replacing each element aij
by the cofactor aij and transposing. Therefore
a11
a
adjoint A = D 21
o
an1
a12
a22
o
an2
p
p
p
a1n T
a2n
T =
o
ann
a11
a
D 12
o
a1n
a21
a22
o
a2n
p
p
p
an1
an2
T.
o
ann
(E.35)
E.5 MATRIX INVERSION
The inverse of a square matrix A is written as A1 and is defined as satisfying the relationship
A1A AA1 I.
(E.36)
adjoint of A
det A
(E.37)
The inverse of a matrix A is
A-1 =
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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Section E.6
Matrices and Characteristic Roots
11
when the det A is not equal to zero. For a 2 2 matrix we have the adjoint matrix
a22
adjoint A = B
-a21
-a12
R,
a11
(E.38)
and the det A a11a22 a12a21. Consider the matrix
1
A = C2
0
2
-1
-1
3
4 S.
1
(E.39)
The determinant has a value det A 7. The cofactor a11 is
a11 = 1-12 2 2
-1
-1
4
2 = 3.
1
(E.40)
In a similar manner we obtain
-1
A
3
adjoint A
1
=
= a- b C -2
det A
7
-2
-5
1
1
11
2 S.
-5
(E.41)
E.6 MATRICES AND CHARACTERISTIC ROOTS
A set of simultaneous linear algebraic equations can be represented by the matrix
equation
y Ax,
(E.42)
where the y vector can be considered as a transformation of the vector x. We might
ask whether it may happen that a vector y may be a scalar multiple of x. Trying
y lx, where l is a scalar, we have
lx Ax.
(E.43)
Alternatively Eq. (E.43) can be written as
lx Ax (lI A)x 0,
(E.44)
where I identity matrix. Thus the solution for x exists if and only if
det (lI A) 0.
(E.45)
This determinant is called the characteristic determinant of A. Expansion of the determinant of Eq. (E.45) results in the characteristic equation. The characteristic equation is an nth-order polynomial in l. The n roots of this characteristic equation are
called the characteristic roots. For every possible value li (i 1, 2, . . . , n) of the nthorder characteristic equation, we can write
(liI A)xi 0.
(E.46)
The vector xi is the characteristic vector for the ith root. Let us consider the matrix
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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Appendix E
An Introduction to Matrix Algebra
2
A= C 2
-1
1
3
-1
1
4 S.
-2
(E.47)
The characteristic equation is found as follows:
1l - 22
det C
-2
1
-1
1l - 32
1
-1
-4 S = 1-l3 + 3l2 + l - 32 = 0.
1l + 22
(E.48)
Ax1 l1x1,
(E.49)
The roots of the characteristic equation are l1 1, l2 1, l3 3. When l l1 1, we find the first characteristic vector from the equation
and we have xT1 k 31 -1
equal to 1. Similarly, we find
04 , where k is an arbitrary constant usually chosen
xT2 30
1
-14,
and
xT3 = 32
3
(E.50)
-14.
E.7 THE CALCULUS OF MATRICES
The derivative of a matrix A A(t) is defined as
d
A1t2 = C
dt
da11 1t2>dt da12 1t2>dt
o
o
dan1 1t2>dt dan2 1t2>dt
p
p
da1n 1t2>dt
o S.
dann 1t2>dt
(E.51)
That is, the derivative of a matrix is simply the derivative of each element aij(t) of the
matrix.
The matrix exponential function is defined as the power series
exp A = eA = I +
A A2 p Ak p
Ak
+
+
+
+
= a
,
1!
2!
k!
k=0 k!
(E.52)
where A2 AA, and, similarly, Ak implies A multiplied k times. This series can be
shown to be convergent for all square matrices. Also a matrix exponential that is a
function of time is defined as
eAt = a
k=0
Aktk
.
k!
(E.53)
If we differentiate with respect to time, then we have
d At
1e 2 = AeAt.
dt
(E.54)
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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
Section E.7
The Calculus of Matrices
13
Therefore for a differential equation
dx
= Ax,
dt
(E.55)
we might postulate a solution x eAtc fc, where the matrix f is f eAt, and c is
an unknown column vector. Then we have
dx
= Ax,
dt
(E.56)
AeAt AeAt,
(E.57)
or
and we have in fact satisfied the relationship, Eq. (E.55). Then the value of c is simply x(0), the initial value of x, because when t 0, we have x(0) c. Therefore the
solution to Eq. (E.55) is
x(t) eAtx(0).
(E.58)
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Modern Control Systems, Eleventh Edition, by Richard C Dorf and Robert H. Bishop.
ISBN: 0132270285. © 2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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