Consider an N x N matrix A with N orthonormal eigenvectors xi such that Axi = λi xi , where the λi is the eigenvalue corrosponding to eigenvector xi . It can be shown that such a matrix A has an expansion of the form : A = (see image) i) Show that if the eigenvalues are real then A , as defined through the above expansion , is Hermitian ii) Using the result for A show that the N x N identify matrix can be witten as : I = (see image) iii) In proving this result for the identify matrix you have used the fact that the vectors {xi} are eigenvectors of a matrix A . Is this essential or is there a milder requirement possible? If so , what is it ? Justify your answers .
Consider an N x N matrix A with N orthonormal eigenvectors xi such that Axi = λi xi , where the λi is the eigenvalue corrosponding to eigenvector xi . It can be shown that such a matrix A has an expansion of the form : A = (see image) i) Show that if the eigenvalues are real then A , as defined through the above expansion , is Hermitian ii) Using the result for A show that the N x N identify matrix can be witten as : I = (see image) iii) In proving this result for the identify matrix you have used the fact that the vectors {xi} are eigenvectors of a matrix A . Is this essential or is there a milder requirement possible? If so , what is it ? Justify your answers .
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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b) Consider an N x N matrix A with N orthonormal eigenvectors xi such that Axi = λi xi , where the λi is the eigenvalue corrosponding to eigenvector xi . It can be shown that such a matrix A has an expansion of the form : A = (see image)
i) Show that if the eigenvalues are real then A , as defined through the above expansion , is Hermitian
ii) Using the result for A show that the N x N identify matrix can be witten as : I = (see image)
iii) In proving this result for the identify matrix you have used the fact that the
Transcribed Image Text:N
N
A = Σ^{x^{x^| = Σλιπ (x).
Α
=
i=1
i=1
Α =
[0 1 07
1 0 1
010
N
1=2x(x).
i=1
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