Tuesday, 4 July 2017

linear algebra - Are singular matrices diagonalizable?




Let An×n be Hermitian with eigenvalues λ1>λ2>>λr=0 and multiplicities q1,...,qr. Can A be diagonalized? Is the matrix of eigenvalues



Ln×n=diag(λ1,,λ1,λ2,,λr1,λr,,λr)



a similar matrix to A?


Answer



Every Hermitian matrix is diagonalizable by the spectral theorem, with its eigenvalues along the diagonal, so the answer to both of your questions is `yes'.


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