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**Unformatted text preview: **for all scalars
x, we would like our deﬁnitions of sin A and cos A to result in the analogous
matrix identity sin2 A + cos2 A = I for all square matrices A. The entrywise
approach (7.3.1) clearly fails in this regard.
One way to deﬁne matrix functions possessing properties consistent with
their scalar counterparts is to use inﬁnite series expansions. For example, consider
the exponential function
∞ ez =
k=0 T
H D
E zk
z2
z3
=1+z+
+
···.
k!
2!
3! IG
R (7.3.2) Formally replacing the scalar argument z by a square matrix A ( z 0 = 1 is
replaced with A0 = I ) results in the inﬁnite series of matrices
e A A2
A3
=I+A+
+
···,
2!
3! (7.3.3) Y
P called the matrix exponential. While this results in a matrix that has properties
analogous to its scalar counterpart, it suﬀers from the fact that convergence must
be dealt with, and then there is the problem of describing the entries in the limit.
These issues are handled by deriving a closed form expression for (7.3.3).
If A is diagonalizable, then A = PDP−1 = P diag (λ1 , . . . , λn ) P−1 , and
k
A = PDk P−1 = P diag λk , . . . , λk P−1 , so
n
1 O
C
∞ eA = k=0 Ak
=
k! ∞ k=0 PDk P−1
=P
k! ∞
k=0 Dk
k! P−1 = P diag eλ1 , . . . , eλn P−1 . In other words, we don’t have to use the inﬁnite series (7.3.3) to deﬁne eA .
Instead, deﬁne eD = diag (eλ1 , eλ2 , . . . , eλn ), and set
eA = PeD P−1 = P diag (eλ1 , eλ2 , . . . , eλn ) P−1 .
This idea can be generalized to any function f (z ) that is deﬁned on the
eigenvalues λi of a diagonalizable matrix A = PDP−1 by deﬁning f (D) to
be f (D) = diag (f (λ1 ), f (λ2 ), . . . , f (λn )) and by setting
f (A) = Pf (D)P−1 = P diag (f (λ1 ), f (λ2 ), . . . , f (λn )) P−1 . Copyright c 2000 SIAM (7.3.4) Buy online from SIAM
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526
Chapter 7
Eigenvalues and Eigenvecto...

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