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matrix exponential∗
mathcam†
2013-03-21 15:52:13
The exponential of a real valued square matrix A, denoted by eA , is defined
as
eA
=
∞
X
1 k
A
k!
k=0
1
= I + A + A2 + · · ·
2
Let us check that eA is a real valued square matrix. Suppose M is a real number
such |Aij | < M for all entries Aij of A. Then |(A2 )ij | < nM 2 for all entries
in A2 , where n is the order of A. (Alternatively, one could argue using matrix
norms: We have ||eA || ≤ e||A|| for the 2-norm, and hence the entries of eA are
bounded by M = ||eA ||.) Thus, in general, we have |(Ak )i,j | < nk M k+1 . Since
P∞ nk k+1
converges, we see that eA converges to real valued n × n matrix.
k=0 k! M
Example 1. Suppose A is nilpotent, i.e., Ar = 0 for some natural number
r. Then
1
1
e A = I + A + A2 + · · · +
Ar−1 .
2!
(r − 1)!
Example 2. If A is diagonalizable, i.e., of the form A = LDL−1 , where D
is a diagonal matrix, then
eA
=
=
∞
X
1
(LDL−1 )k
k!
k=0
∞
X
1
LDk L−1
k!
k=0
D
= Le L−1 .
Further, if D = diag{a1 , · · · , an }, then Dk = diag{ak1 , · · · , akn } whence
eA
= L diag{ea1 , · · · , ean }L−1 .
∗ hMatrixExponentiali
created: h2013-03-21i by: hmathcami version: h34162i Privacy
setting: h1i hDefinitioni h15A15i h15-00i
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For diagonalizable matrix A, it follows that det eA = etr A . However, this formula is, in fact, valid for all A.
Properties
Let A be a square n×n real valued matrix. Then the matrix exponential satisfies
the following properties
1. For the n × n zero matrix O, eO = I, where I is the n × n identity matrix.
2. If A = L diag{a1 , · · · , an }L−1 for an invertible n × n matrix L, then
eA = L diag{ea1 , · · · , ean }L−1 .
3. If A and B commute, then eA+B = eA eB .
4. The trace of A and the determinant of eA are related by the formula
det eA = etr A .
In effect, eA is always invertible. The inverse is given by
(eA )−1 = e−A .
5. If eA is a rotational matrix, then tr A = 0.
A relevant example on property 3.
We report an interesting example where the cited property is valid. In the field
of complex numbers consider the complex matrix
C = A + iB,
(1)
being C hermitian, i.e. C | = C̄ (here ”|” and overline ”−” stand for tranposition and conjugation, respectively) and orthogonal, i.e C −1 = C | . From
(1),
C | = A| + iB | .
Since C is orthogonal, from the complex equation CC | = I (I is the identity
matrix), we have
CC | = (A + iB)(A| + iB | ) = (AA| − BB | ) + i(BA| + AB | ) = I,
whence the imaginary part leads to the equation
BA| + AB | = 0.
(2)
But C is also hermitian, so that
C | = A| + iB | = C̄ = A − iB,
therefore A| = A is symmetric, and B | = −B is skew-symmetric. From these
and (2), BA = AB, and this implies that exp(A)·exp(B) = exp(A+B). So that,
the real and imaginary parts of an orthogonal and hermitian matrix verifies the
property. Likewise, it is easy to show that if the complex matrix is symmetric
and unitary, its real an imaginary components also verify this property.
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