MATH307 April 2010
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Question 05 (a)
Consider a matrix , a real diagonal matrix containing the eigenvalues of (all positive) and a matrix V that diagonalizes . Answer the following questions, substantiate all your statements.
Explain why exists and show that the matrix is unitary.
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Let . Since contains the eigenvalues on its diagonal, S would be the matrix with on the diagonal. Since , is invertible as .
Since is Hermitian, it can be diagonalized with a unitary matrix and let .
... and since ,
And thus, U is unitary.