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Consider the matrix \(N = \begin{bmatrix} 1 & \mathrm{i} \\ \mathrm{i} & 1 \end{bmatrix} \).
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Verify that \(N\) is normal. Is \(N\) Hermitian?
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Determine a unitary \(U\) so that \(U^\ast N U\) is diagonal.
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Suppose \(\theta\) is a fixed angle with \(0 < \theta < 2\pi\),
and set \(N = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \).
Even though the entries of \(N\) are real, regard it as a complex matrix.
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Verify that \(N\) is normal.
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Verify that the eigenvalues of \(N\) are \(e^{\mathrm{i} \theta}\) and \(e^{- \mathrm{i} \theta}\).
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Determine a unitary \(U\) so that
\(U^\ast N U = \begin{bmatrix} e^{\mathrm{i} \theta} & 0 \\ 0 & e^{- \mathrm{i} \theta} \end{bmatrix} \).
Hint You may find the circle identity useful: \(\sin^2 \theta + \cos^2 \theta = 1\).
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Consider the square matrix
\( N = \left[\begin{smallmatrix} N_1 \\ & N_2 \end{smallmatrix}\right] \)
in block-diagonal form.
Prove that if the blocks \(N_1,N_2\) are both normal, then so is \(N\).
(Refer to problem #2(a) from the page of extra practise problems for matrix adjoints.)