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A = To 0 1 10 010 0 1 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=[x'(x¹)¹. iii) In proving this result for the identity matrix you have used the fact that the vectors

A = To 0 1 10 010 0 1 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=[x'(x¹)¹. iii) In proving this result for the identity matrix you have used the fact that the vectors

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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A = To 1 0 10 1 010 Consider an Nx N matrix A with N orthonormal eigenvectors x' such that Ax'=A₁x¹, 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: N A=A|x) (x²|=A‚x²(x²)¹. 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 -Σx²(x²)¹. i=1 I= 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. iv) Verify the above identity for I using the eigenvectors of A from the first part of this If you have been unable to find suitable eigenvectors of the A, then use the following set for this part of the question: x¹ = (0, 0, 1), x² = (1/√2) (1, 1, 0)", and x³ = (1/√2) (1, 1, 0)T. Note, these are not the eigenvectors of A.
Transcribed Image Text:A = To 1 0 10 1 010 Consider an Nx N matrix A with N orthonormal eigenvectors x' such that Ax'=A₁x¹, 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: N A=A|x) (x²|=A‚x²(x²)¹. 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 -Σx²(x²)¹. i=1 I= 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. iv) Verify the above identity for I using the eigenvectors of A from the first part of this If you have been unable to find suitable eigenvectors of the A, then use the following set for this part of the question: x¹ = (0, 0, 1), x² = (1/√2) (1, 1, 0)", and x³ = (1/√2) (1, 1, 0)T. Note, these are not the eigenvectors of A.
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