Math Problems Vault

Problem 6.2.1

(i)(ii)

The star product of the Hadamard matrix $U_{H}$ with itself provides the Bell matrix

B = U_{H} \star U_{H} = \frac{1}{\sqrt{2}} \left(\begin{smallmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{smallmatrix}\right)

which is a unitary matrix. Note that the eigenvalues of the Bell matrix are $+1$ (twice) and $-1$ (twice).

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(i)

Find the square roots of the Bell matrix applying the spectral theorem.

(ii)

Apply the spectral theorem to find the skew-hermitian matrix $K$ such that $B = e^{K}$ .

Problem 6.2.2

Let $A$ be a normal matrix with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ and pairwise orthonormal eigenvectors $\overrightarrow {a}_{j}$ (column vectors). Let $B$ be a normal matrix with eigenvalues $\mu_{1}, \ldots , \mu_{n}$ and pairwise orthonormal eigenvectors $\overrightarrow {b}_{j}$ (column vectors). Then $A$ and $B$ can be written as

A = \sum _{j=1}^{n} \lambda _{j}\overrightarrow {a}_{j}\overrightarrow {a}_{j}^{*}, \qquad B = \sum _{k=1}^{n} \mu _{k}\overrightarrow {b}_{k}\overrightarrow {b}_{k}^{*}.

Let $z \in \mathbb {C}$ . Use the spectral decomposition to calculate $e^{zA}Be^{-zA}$ .

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Problem 6.2.3

Consider the normalized vectors

\overrightarrow {v}_{1} = \frac{1}{\sqrt{2}}\left(\begin{smallmatrix} 1 \\ 0 \\ 1 \end{smallmatrix}\right), \quad \overrightarrow {v}_{2} = \left(\begin{smallmatrix} 0 \\ 1 \\ 0 \end{smallmatrix}\right), \quad \overrightarrow {v}_{3} = \frac{1}{\sqrt{2}}\left(\begin{smallmatrix} 1 \\ 0 \\ -1 \end{smallmatrix}\right)

in $\mathbb {R}^{3}$ . Find the eigenvalues and eigenvectors of the $3 \times 3$ matrix $M = 1 \cdot \overrightarrow {v}_{1}\overrightarrow {v}_{1}^{*} + 0 \cdot \overrightarrow {v}_{2}\overrightarrow {v}_{2}^{*} - 1 \cdot \overrightarrow {v}_{3}\overrightarrow {v}_{3}^{*}$ without calculating $M$ .

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Problem 6.2.4

Let $A$ be an $n \times n$ normal matrix with eigenvalues $\lambda_{1}, \ldots , \lambda_{n}$ and pairwise normalized orthogonal eigenvectors $\overrightarrow {v}_{j}$ ( $j = 1, \ldots , n$ ). Then

A = \sum _{j=1}^{n} \lambda _{j}\overrightarrow {v}_{j}\overrightarrow {v}_{j}^{*}.

Show that $\cos (A) = \sum_{j=1}^{n} \cos (\lambda_{j})\overrightarrow {v}_{j}\overrightarrow {v}_{j}^{*}$ , $\sin (A) = \sum_{j=1}^{n} \sin (\lambda_{j})\overrightarrow {v}_{j}\overrightarrow {v}_{j}^{*}$ .

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Problem 6.2.5

The spectral theorem for $n \times n$ normal matrices over $\mathbb {C}$ is as follows: A matrix $A$ is normal if and only if there exists an $n \times n$ unitary matrix $U$ and a diagonal matrix $D$ such that $D = U^{*}AU$ . Use this theorem to prove that the matrix

A = \left(\begin{smallmatrix} 0 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{smallmatrix}\right)

is nonnormal.

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Problem 6.2.6

Find all $2 \times 2$ matrices over $\mathbb {C}$ which admit the normalized eigenvectors

\overrightarrow {v}_{1} = \frac{1}{\sqrt{2}}\left(\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right), \qquad \overrightarrow {v}_{2} = \frac{1}{\sqrt{2}}\left(\begin{smallmatrix} 1 \\ -1 \end{smallmatrix}\right)

with the corresponding eigenvalues $\lambda_{1}$ and $\lambda_{2}$ .

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Problem 6.2.7

Consider the nonnormal $2 \times 2$ matrix

A = \left(\begin{smallmatrix} 1 & 1 \\ 0 & -1 \end{smallmatrix}\right)

with eigenvalues $\lambda_{1} = +1$ and $\lambda_{2} = -1$ and normalized eigenvectors

\overrightarrow {v}_{1} = \left(\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right), \qquad \overrightarrow {v}_{2} = \frac{2}{\sqrt{5}}\left(\begin{smallmatrix} 1/2 \\ -1 \end{smallmatrix}\right).

The eigenvectors are linearly independent, but not orthonormal. Calculate $\overrightarrow {v}_{1}^{*}\overrightarrow {v}_{2}$ and

B = \lambda _{1}\overrightarrow {v}_{1}\overrightarrow {v}_{1}^{*} + \lambda _{2}\overrightarrow {v}_{2}\overrightarrow {v}_{2}^{*}.

Find $\left\| A - B\right\|$ . Discuss. Is the matrix $B$ normal?

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