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(b) Explain why we must have that det(S – A,I) = 0 for each of the eigenvalues of S, where I is the identity matrix.

(b) Explain why we must have that det(S – A,I) = 0 for each of the eigenvalues of S, where I is the identity matrix.

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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From practice sheet I need the solution To b)

Let S be a symmetric matrix, and the set of solutions of the equation Sen = Anen define the eigenvalues A, and eigenvectors 2, of S. (a) Suppose that S is written in the form S = 0" DO, where O is an orthogonal matrix and D is a diagonal matrix. State how the structures of O and D are related to the eigenvalues and eigenvectors of S. (b) Explain why we must have that det(S – A,I) = 0 for each of the eigenvalues of S, where I is the identity matrix.
Transcribed Image Text:Let S be a symmetric matrix, and the set of solutions of the equation Sen = Anen define the eigenvalues A, and eigenvectors 2, of S. (a) Suppose that S is written in the form S = 0" DO, where O is an orthogonal matrix and D is a diagonal matrix. State how the structures of O and D are related to the eigenvalues and eigenvectors of S. (b) Explain why we must have that det(S – A,I) = 0 for each of the eigenvalues of S, where I is the identity matrix.
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