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Let A be an n x n symmetric matrix. Let {f₁,...,fn} be an orthonormal basis of eigenvectors of A satisfying Afj=Xjf, for j = 1,..., n. - ΣΑ,1,17. j=1 (a) Show that A = (This formula is called the spectral decomposition of A.) (Hint: compare the (i, j)-entry of Aff with the (i, j)-entry of A = PDPT.) j=1 (b) Given polynomial p(x) = akr +ak-12-¹+...+ a₁x + ao, define the matrix p(A) = akA* +ak-1A¹+...+a₁A+aon. Show that p(X1),...,p(An) are the eigenvalues of p(A) and p(A) = Σv(x,)f,ff. 22 j-1

Let A be an n x n symmetric matrix. Let {f₁,...,fn} be an orthonormal basis of eigenvectors of A satisfying Afj=Xjf, for j = 1,..., n. - ΣΑ,1,17. j=1 (a) Show that A = (This formula is called the spectral decomposition of A.) (Hint: compare the (i, j)-entry of Aff with the (i, j)-entry of A = PDPT.) j=1 (b) Given polynomial p(x) = akr +ak-12-¹+...+ a₁x + ao, define the matrix p(A) = akA* +ak-1A¹+...+a₁A+aon. Show that p(X1),...,p(An) are the eigenvalues of p(A) and p(A) = Σv(x,)f,ff. 22 j-1

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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Let A be an n x n symmetric matrix. Let {f₁,...,fn} be an orthonormal basis of eigenvectors of A satisfying Afj=Xjfj, for j = 1,..., n. (a) Show that A = 12 =ΣALET. j=1 (This formula is called the spectral decomposition of A.) (Hint: compare the (i, j)-entry of Aff with the (i, j)-entry of A = PDPT.) j=1 (b) Given polynomial p(x) = akr +ak-12-¹+...+ a₁x + ao, define the matrix p(A) = akA" +ak- ak-1A jk-1 +...+ a₁A+aoIn. Show that p(X1),...,p(An) are the eigenvalues of p(A) and 22 p(A) = Σp(x)fff. j=1 This question implies CA (A) = 0 if сл(x) (x - ₁)(x − λ₂) (An). Your proof leads to the Cayley-Hamilton Theorem for symmetric matrices. Cayley-Hamilton Theorem: Let CA(z) be the characteristic polynomial of a square matrix A. Then CA(A) is the zero matrix.
Transcribed Image Text:Let A be an n x n symmetric matrix. Let {f₁,...,fn} be an orthonormal basis of eigenvectors of A satisfying Afj=Xjfj, for j = 1,..., n. (a) Show that A = 12 =ΣALET. j=1 (This formula is called the spectral decomposition of A.) (Hint: compare the (i, j)-entry of Aff with the (i, j)-entry of A = PDPT.) j=1 (b) Given polynomial p(x) = akr +ak-12-¹+...+ a₁x + ao, define the matrix p(A) = akA" +ak- ak-1A jk-1 +...+ a₁A+aoIn. Show that p(X1),...,p(An) are the eigenvalues of p(A) and 22 p(A) = Σp(x)fff. j=1 This question implies CA (A) = 0 if сл(x) (x - ₁)(x − λ₂) (An). Your proof leads to the Cayley-Hamilton Theorem for symmetric matrices. Cayley-Hamilton Theorem: Let CA(z) be the characteristic polynomial of a square matrix A. Then CA(A) is the zero matrix.
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