Consider an Nx N matrix A with N orthonormal eigenvectors x such that Ax' = ₁x¹, where the X, is the eigenvalue corresponding to eigenvector x'. It can be shown that such a matrix A has an expansion of the form: A = ΣA₁x¹)(x¹=A₁x²(x¹). 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 Nx N identity matrix can be written as N I= Σx'(x¹)¹. iii) In proving this result for the identity matrix you have used the fact that the vectors {x} 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 Nx N matrix A with N orthonormal eigenvectors x such that Ax' = ₁x¹, where the X, is the eigenvalue corresponding to eigenvector x'. It can be shown that such a matrix A has an expansion of the form: A = ΣA₁x¹)(x¹=A₁x²(x¹). 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 Nx N identity matrix can be written as N I= Σx'(x¹)¹. iii) In proving this result for the identity matrix you have used the fact that the vectors {x} 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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Transcribed Image Text:0 1 0
A = 1 0
010
Transcribed Image Text:Consider an Nx N matrix A with N orthonormal eigenvectors x such that Ax' = ₁x¹,
where the X, is the eigenvalue corresponding to eigenvector x'. It can be shown that such
a matrix A has an expansion of the form:
A = ΣA₁x¹)(x¹=A₁x²(x¹).
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 Nx N identity matrix can be written as
N
I= Σx'(x¹)¹.
iii) In proving this result for the identity matrix you have used the fact that the vectors
{x} are eigenvectors of a matrix A. Is this essential or is there a milder requirement
possible? If so, what is it? Justify your answers.
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