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a) Consider a matrix A with N orthonormal eigenvectors {x'} and eigenvalues {A,}. Construct the matrix S from the eigenvectors of A as follows: s- [x' x x"] where each column of S is an eigenvector of A. Because S is constructed from a set of linearly independent vectors, the inverse matrix S-' exists. I) State general conditions on A for it to have a set of N orthonormal eigenvectors. i) Show that S is unitary, i.e., that S' = s-!. li) Show that the matrix A' = S'AS is diagonal, and A', = A,6,y.

a) Consider a matrix A with N orthonormal eigenvectors {x'} and eigenvalues {A,}. Construct the matrix S from the eigenvectors of A as follows: s- [x' x x"] where each column of S is an eigenvector of A. Because S is constructed from a set of linearly independent vectors, the inverse matrix S-' exists. I) State general conditions on A for it to have a set of N orthonormal eigenvectors. i) Show that S is unitary, i.e., that S' = s-!. li) Show that the matrix A' = S'AS is diagonal, and A', = A,6,y.

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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a) Consider a matrix A with N orthonormal eigenvectors {x'} and eigenvalues {A,}. Construct the matrix S from the eigenvectors of A as follows: s- [x' x? .. x") where each column of S is an eigenvector of A. Because S is constructed from a set of linearly independent vectors, the inverse matrix S-' exists. i) State general conditions on A for it to have a set of N orthonormal eigenvectors. ii) Show that S is unitary, i.e., that S' =S-!. i) Show that the matrix A' = S'AS is diagonal, and A', = A,6,y. iv) Now consider the action of A" on a general vector y in this N dimensional space. We wish to evaluate A"y. It is proposed that we start by expanding y as y = E 4,x'. Answer the following: 1. Explain why y can be expanded as shown. Quote relevant theorems/results. 2. Find an expression for the coefficients a, in terms of a suitable inner product. 3. Show that A"y =E, \"a,x'.
Transcribed Image Text:a) Consider a matrix A with N orthonormal eigenvectors {x'} and eigenvalues {A,}. Construct the matrix S from the eigenvectors of A as follows: s- [x' x? .. x") where each column of S is an eigenvector of A. Because S is constructed from a set of linearly independent vectors, the inverse matrix S-' exists. i) State general conditions on A for it to have a set of N orthonormal eigenvectors. ii) Show that S is unitary, i.e., that S' =S-!. i) Show that the matrix A' = S'AS is diagonal, and A', = A,6,y. iv) Now consider the action of A" on a general vector y in this N dimensional space. We wish to evaluate A"y. It is proposed that we start by expanding y as y = E 4,x'. Answer the following: 1. Explain why y can be expanded as shown. Quote relevant theorems/results. 2. Find an expression for the coefficients a, in terms of a suitable inner product. 3. Show that A"y =E, \"a,x'.
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