Let A be an n x n matrix over C. Let ƒ be an entire function, i.e. an analytic function on the whole complex plane, for example exp(z), sin(z), cos(z). Using the Cayley-Hamilton theorem we can write ƒ(A) = an−1A”−¹ + an−2A²−²+...+ a₂ò + a1A+aoIn (1) where the complex numbers ao, a₁, ..., an-1 are determined as follows: Let r(A) := ªn−1λ²−¹ + an-2√²-2 + ... + a₂λ² + a₁λ +ao which is the right-hand side of (1) with A³ replaced by X³, where j = 0, 1, …, n— 1 of each distinct eigenvalue λ; of the matrix A, we consider the equation ƒ(λj)=r(λj). (2) If A, is an eigenvalue of multiplicity k, for k> 1, then we consider also the following equations ƒ'(A)|a=a; = r′(A)|a=a; ƒ"(A)|a=a; = 5″(A)|a=a; ...... f (k-1)(x)=(-1) (X)|₁x)" (x)}|x=x; * Apply this technique to find exp(A) with с ^= (ε c), A CER, C# 0. C
Let A be an n x n matrix over C. Let ƒ be an entire function, i.e. an analytic function on the whole complex plane, for example exp(z), sin(z), cos(z). Using the Cayley-Hamilton theorem we can write ƒ(A) = an−1A”−¹ + an−2A²−²+...+ a₂ò + a1A+aoIn (1) where the complex numbers ao, a₁, ..., an-1 are determined as follows: Let r(A) := ªn−1λ²−¹ + an-2√²-2 + ... + a₂λ² + a₁λ +ao which is the right-hand side of (1) with A³ replaced by X³, where j = 0, 1, …, n— 1 of each distinct eigenvalue λ; of the matrix A, we consider the equation ƒ(λj)=r(λj). (2) If A, is an eigenvalue of multiplicity k, for k> 1, then we consider also the following equations ƒ'(A)|a=a; = r′(A)|a=a; ƒ"(A)|a=a; = 5″(A)|a=a; ...... f (k-1)(x)=(-1) (X)|₁x)" (x)}|x=x; * Apply this technique to find exp(A) with с ^= (ε c), A CER, C# 0. C
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Let A be an n x n matrix over C. Let f be an entire function,
i.e. an analytic function on the whole complex plane, for example exp(z), sin(z),
cos(z). Using the Cayley-Hamilton theorem we can write
ƒ(A) = an−1A”−¹ + an−2A¹−²+...+ a₂ò + a₁A+aoIn (1)
where the complex numbers ao, a₁, ..., an-1 are determined as follows: Let
r(λ) := An−1\”−¹ + an-21²-2+...+ a₂λ² + a₁λ + ao
which is the right-hand side of (1) with A¹ replaced by X³, where j = 0, 1, ….., n—
1 of each distinct eigenvalue λ; of the matrix A, we consider the equation
ƒ(λj) =r(λj).
(2)
If X, is an eigenvalue of multiplicity k, for k> 1, then we consider also the
following equations
ƒ'(A)|a=a; = r′(A)|a=a;
ƒ"(A)|a=a; = r″(A)|a=a;
......
f(k-1) (X) = (k-1)(A)
(A)|x
1x=x₂
Apply this technique to find exp(A) with
с
A =
= (ε c),
C
C
*
|x=x₂
CER, C# 0.
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