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[ 8.88178420e-16 4.44089210e-16 2.66453526e-15]]
>>> print abs(A-Z1*T1*Z1.H) # same
[[ 1.00043248e-15 2.22301403e-15 5.55749485e-15]
[ 2.88899660e-15 8.44927041e-15 9.77322008e-15]
[ 3.11291538e-15 1.15463228e-14 1.15464861e-14]]
>>> print abs(A-Z2*T2*Z2.H) # same
[[ 3.34058710e-16 8.88611201e-16 4.18773089e-18]
[ 1.48694940e-16 8.95109973e-16 8.92966151e-16]
[ 1.33228956e-15 1.33582317e-15 3.55373104e-15]]
1.8.4 Matrix Functions
Consider the function f (x) with Taylor series expansion
f (x) =
∞�
k=0
f (k) (0)
x
k!
k .
A matrix function can be defined using this Taylor series for the square matrix A as
f (A) =
∞�
k=0
f (k) (0)
k! Ak .
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While, this serves as a useful representation of a matrix function, it is rarely the best way to calculate a matrix function.
Exponential and logarithm functions
The matrix exponential is one of the more common matrix functions. It can be defined for square matrices as
e A =
∞�
k=0
1
k! Ak .
The command linalg.expm3 uses this Taylor series definition to compute the matrix exponential. Due to poor
convergence properties it is not often used.
Another method to compute the matrix exponential is to find an eigenvalue decomposition of A :
and note that
A = VΛV −1
e A = Ve Λ V −1
where the matrix exponential of the diagonal matrix Λ is just the exponential of its elements. This method is implemented
in linalg.expm2 .
The preferred method for implementing the matrix exponential is to use scaling and a Padé approximation for e x .
This algorithm is implemented as linalg.expm .
The inverse of the matrix exponential is the matrix logarithm defined as the inverse of the matrix exponential.
A ≡ exp (log (A)) .
The matrix logarithm can be obtained with linalg.logm .
1.8. Linear Algebra 47