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
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'.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66578098-c5d5-436d-b59c-5cded03aa021%2F4a139620-77a0-4b1e-98b7-a025086525ee%2Fytqyf3_processed.png&w=3840&q=75)
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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