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A = Го 1 0 101 10 Lo 0 Consider an Nx N matrix A with N orthonormal eigenvectors x' such that Ax' = x¹, 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=ΣA|x) (x¹=[A₁x²(x¹)¹. -£Ax(x) 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 1=Σx'(x¹)¹. i=1

A = Го 1 0 101 10 Lo 0 Consider an Nx N matrix A with N orthonormal eigenvectors x' such that Ax' = x¹, 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=ΣA|x) (x¹=[A₁x²(x¹)¹. -£Ax(x) 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 1=Σx'(x¹)¹. i=1

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 = To 1 0 101 010 Consider an Nx N matrix A with N orthonormal eigenvectors x' such that Ax' = x¹, where the A, is the eigenvalue corresponding to eigenvector x'. It can be shown that such a matrix A has an expansion of the form: A =Σ/x)(x| = Σ\x(x)". 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 1=Σx'(x¹)¹. i=1
Transcribed Image Text:A = To 1 0 101 010 Consider an Nx N matrix A with N orthonormal eigenvectors x' such that Ax' = x¹, where the A, is the eigenvalue corresponding to eigenvector x'. It can be shown that such a matrix A has an expansion of the form: A =Σ/x)(x| = Σ\x(x)". 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 1=Σx'(x¹)¹. i=1
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