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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

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 32E
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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.
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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