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Exercise 7.2.7 Suppose A is an nxn matrix and let V be an eigenvector such that AV = V. Also suppose the characteristic polynomial of A is det (xI-A) = x + an-1xh ++a₁x+ao Explain why (A” + an-1A¹-1. +...+a₁A+aol) V = 0 If A is diagonalizable, give a proof of the Cayley Hamilton theorem based on this. This theorem says A satisfies its characteristic equation, A"+an-1A" |n-1 ++a₁A+aol = 0

Exercise 7.2.7 Suppose A is an nxn matrix and let V be an eigenvector such that AV = V. Also suppose the characteristic polynomial of A is det (xI-A) = x + an-1xh ++a₁x+ao Explain why (A” + an-1A¹-1. +...+a₁A+aol) V = 0 If A is diagonalizable, give a proof of the Cayley Hamilton theorem based on this. This theorem says A satisfies its characteristic equation, A"+an-1A" |n-1 ++a₁A+aol = 0

Advanced Engineering Mathematics
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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Exercise 7.2.7 Suppose A is an nxn matrix and let V be an eigenvector such that AV = V. Also suppose the characteristic polynomial of A is det (x1 - A)=x"+an_1x². ++a₁x+ao Explain why (A"+an-1An-1 ++ a₁A+aol) V = 0 If A is diagonalizable, give a proof of the Cayley Hamilton theorem based on this. This theorem says A satisfies its characteristic equation, A" +an-14"-1. ++a₁A+aol = 0
Transcribed Image Text:Exercise 7.2.7 Suppose A is an nxn matrix and let V be an eigenvector such that AV = V. Also suppose the characteristic polynomial of A is det (x1 - A)=x"+an_1x². ++a₁x+ao Explain why (A"+an-1An-1 ++ a₁A+aol) V = 0 If A is diagonalizable, give a proof of the Cayley Hamilton theorem based on this. This theorem says A satisfies its characteristic equation, A" +an-14"-1. ++a₁A+aol = 0
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