Math Problems Vault

Problem 8.2.1

If $A \in \mathbb {R}^{n \times n}$ , then there exists an orthogonal $Q \in \mathbb {R}^{n \times n}$ such that

Q^{T}AQ = \begin{pmatrix} R_{11} & R_{12} & & R_{1m} \\ 0 & R_{22} & & R_{2m} \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \ldots & R_{mn} \end{pmatrix}

where each $R_{ii}$ is either a $1 \times 1$ matrix or a $2 \times 2$ matrix having complex conjugate eigenvalues. Find $Q$ for the $3 \times 3$ matrix

A = \begin{pmatrix} 0 & 1 & 0 \\ 2 & 0 & 3 \\ 0 & 4 & 0 \end{pmatrix} .

Then calculate $Q^{T}AQ$ .

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Problem 8.2.2

Find a cosine-sine decomposition of the matrices

\frac{1}{\sqrt{2}} \begin{pmatrix} i & i \\ i & -i \end{pmatrix} , \qquad \frac{1}{\sqrt{2}} \begin{pmatrix} i & i \\ i & -i \end{pmatrix} \otimes \frac{1}{\sqrt{2}} \begin{pmatrix} i & i \\ i & -i \end{pmatrix} .

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Problem 8.2.3

(i)(ii)

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(i)

Consider the $4 \times 4$ matrices

\Omega = \begin{pmatrix} 0 & 1 & 0 & 0 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & 0 \end{pmatrix} , \qquad \widetilde{\Omega } = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \end{pmatrix} .

Can one find $4 \times 4$ permutation matrices $P, Q$ such that $\Omega = P \widetilde{\Omega } Q$ ?

(ii)

Consider the $2n \times 2n$ matrices

\Omega = \begin{pmatrix} 0 & 1 & 0 & \ldots & 0 & 0 & 0 \\ -1 & 0 & 1 & \ldots & 0 & 0 & \\ 0 & -1 & 0 & \ldots & 0 & 0 & 0 \\ & & & \ddots & & & \\ 0 & 0 & 0 & & -1 & 0 & 1 \\ 0 & 0 & 0 & & 0 & -1 & 0 \end{pmatrix} , \qquad \widetilde{\Omega } = \begin{pmatrix} 0_{n} & I_{n} \\ -I_{n} & 0_{n} \end{pmatrix} .

Can one find $2n \times 2n$ permutation matrices $P, Q$ such that $\Omega = P \widetilde{\Omega } Q$ ?

Problem 8.2.4

(i)(ii)

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(i)

Given the $3 \times 2$ matrix

A = \begin{pmatrix} 0 & 1 \\ 1 & 1 \\ 1 & 0 \end{pmatrix} .

Find the singular value decomposition of $A$ .

(ii)

Let $\alpha \in \mathbb {R}$ . Find the singular value decomposition of the $2 \times 3$ matrix

X(\alpha ) = \begin{pmatrix} \cos (\alpha ) & \sin (\alpha ) & 0 \\ 0 & \cos (\alpha ) & \sin (\alpha ) \end{pmatrix} .

Problem 8.2.5

The Cholesky decomposition of the a positive semi-definite matrix $M$ is the decomposition into a product of a lower triangular matrix $L$ and its conjugate transpose $L^{*}$ , i.e. $M = LL^{*}$ . Find the Cholesky decomposition for the density matrix (pure state)

\rho = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 \\ -i \\ i \end{pmatrix} \frac{1}{\sqrt{3}} (1 \ \ i \ \ -i) \equiv \frac{1}{3} \begin{pmatrix} 1 & i & -i \\ -i & 1 & -1 \\ i & -1 & 1 \end{pmatrix} .

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Problem 8.2.6

Show that (cosine-sine decomposition)

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} i & 0 \\ 0 & 1 \end{pmatrix} .

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Problem 8.2.7

Let $U$ be an $n \times n$ unitary matrix. The matrix $U$ can always be diagonalized by a unitary matrix $V$ such that

U = V \begin{pmatrix} e^{i\theta _{1}} & & 0 \\ & \vdots & \\ 0 & & e^{i\theta _{n}} \end{pmatrix} V^{*}

where $e^{i\theta_{j}}$ , $\theta_{j} \in [0, 2\pi )$ are the eigenvalues of $U$ . Let $n = 2$ and

U = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} .

Thus the eigenvalues of $U$ are 1 and $-1$ . Show that

U = V \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} V^{*} \quad \text{with} \quad V = V^{*} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} .

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Problem 8.2.8

If $A \in SL(2, \mathbb {R})$ , then it can be uniquely be written in the form

\begin{pmatrix} \cos (\phi ) & \sin (\phi ) \\ -\sin (\phi ) & \cos (\phi ) \end{pmatrix} \exp \begin{pmatrix} a & b \\ b & -a \end{pmatrix} .

Find this decomposition for the $2 \times 2$ nonnormal matrix

A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} .

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