Math Problems Vault

Problem 3.6.1

For the partitioned matrices

\mathbf{A} = \left[\begin{array}{c|cc|ccc} 1 & 0 & 0 & 3 & 3 & 3 \\ 1 & 0 & 0 & 3 & 3 & 3 \\ \hline 1 & 2 & 2 & 0 & 0 & 0 \end{array}\right] \qquad \text{and} \qquad \mathbf{B} = \left[\begin{array}{cc} -1 & -1 \\ \hline 0 & 0 \\ 0 & 0 \\ \hline -1 & -2 \\ -1 & -2 \\ -1 & -2 \end{array}\right],

use block multiplication with the indicated partitions to form the product $\mathbf{A}\mathbf{B}$ .

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Problem 3.6.2

For all matrices $\mathbf{A}_{n \times k}$ and $\mathbf{B}_{k \times n}$ , show that the block matrix

\mathbf{L} = \begin{pmatrix} \mathbf{I}-\mathbf{B}\mathbf{A} & \mathbf{B} \\ 2\mathbf{A}-\mathbf{A}\mathbf{B}\mathbf{A} & \mathbf{A}\mathbf{B}-\mathbf{I} \end{pmatrix}

has the property $\mathbf{L}^{2} = \mathbf{I}$ . Matrices with this property are said to be involutory, and they occur in the science of cryptography.

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Problem 3.6.3

For the matrix

\mathbf{A} = \begin{bmatrix} 1 & 0 & 0 & 1/3 & 1/3 & 1/3 \\ 0 & 1 & 0 & 1/3 & 1/3 & 1/3 \\ 0 & 0 & 1 & 1/3 & 1/3 & 1/3 \\ 0 & 0 & 0 & 1/3 & 1/3 & 1/3 \\ 0 & 0 & 0 & 1/3 & 1/3 & 1/3 \\ 0 & 0 & 0 & 1/3 & 1/3 & 1/3 \end{bmatrix},

determine $\mathbf{A}^{300}$ .

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Problem 3.6.4

For every matrix $\mathbf{A}_{m \times n}$ , demonstrate that the products $\mathbf{A}^{*}\mathbf{A}$ and $\mathbf{A}\mathbf{A}^{*}$ are hermitian matrices.

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Problem 3.6.5

If $\mathbf{A}$ and $\mathbf{B}$ are symmetric matrices that commute, prove that the product $\mathbf{A}\mathbf{B}$ is also symmetric. If $\mathbf{A}\mathbf{B} \neq \mathbf{B}\mathbf{A}$ , is $\mathbf{A}\mathbf{B}$ necessarily symmetric?

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Problem 3.6.6

Prove that the right-hand distributive property is true: for conformable matrices $\mathbf{D}$ , $\mathbf{E}$ , $\mathbf{F}$ , $(\mathbf{D}+\mathbf{E})\mathbf{F} = \mathbf{D}\mathbf{F}+\mathbf{E}\mathbf{F}$ .

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Problem 3.6.7

For each matrix $\mathbf{A}_{n \times n}$ , explain why it is impossible to find a solution for $\mathbf{X}_{n \times n}$ in the matrix equation

\mathbf{A}\mathbf{X}-\mathbf{X}\mathbf{A} = \mathbf{I}.

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Problem 3.6.8

(a)(b)

Let $\overrightarrow {y}^{T}_{1 \times m}$ be a row of unknowns, and let $\mathbf{A}_{m \times n}$ and $\overrightarrow {b}^{T}_{1 \times n}$ be known matrices.

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(a)

Explain why the matrix equation $\overrightarrow {y}^{T}\mathbf{A} = \overrightarrow {b}^{T}$ represents a system of $n$ linear equations in $m$ unknowns.

(b)

How are the solutions for $\overrightarrow {y}^{T}$ in $\overrightarrow {y}^{T}\mathbf{A} = \overrightarrow {b}^{T}$ related to the solutions for $\overrightarrow {x}$ in $\mathbf{A}^{T}\overrightarrow {x} = \overrightarrow {b}$ ?

Problem 3.6.9

(a)(b)

A particular electronic device consists of a collection of switching circuits that can be either in an ON state or an OFF state. These electronic switches are allowed to change state at regular time intervals called clock cycles. Suppose that at the end of each clock cycle, $30\%$ of the switches currently in the OFF state change to ON, while $90\%$ of those in the ON state revert to OFF.

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(a)

Show that the device approaches an equilibrium in the sense that the proportion of switches in each state eventually becomes constant, and determine these equilibrium proportions.

(b)

Independent of the initial proportions, about how many clock cycles does it take for the device to become essentially stable?

Problem 3.6.10

Recall Example 3.6.7's reducibility technique: if $\mathbf{T}_{n \times n}\overrightarrow {x} = \overrightarrow {b}$ has a block-triangular coefficient matrix $\mathbf{T} = \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{0} & \mathbf{C} \end{pmatrix}$ (with $\mathbf{A}$ being $r \times r$ and $\mathbf{C}$ being $(n-r) \times (n-r)$ ), and $\overrightarrow {x}$ , $\overrightarrow {b}$ are partitioned to match as $\overrightarrow {x} = \begin{pmatrix} \overrightarrow {x}_{1} \\ \overrightarrow {x}_{2} \end{pmatrix}$ , $\overrightarrow {b} = \begin{pmatrix} \overrightarrow {b}_{1} \\ \overrightarrow {b}_{2} \end{pmatrix}$ , then block multiplication reduces $\mathbf{T}\overrightarrow {x} = \overrightarrow {b}$ to two smaller systems, $\mathbf{A}\overrightarrow {x}_{1}+\mathbf{B}\overrightarrow {x}_{2} = \overrightarrow {b}_{1}$ and $\mathbf{C}\overrightarrow {x}_{2} = \overrightarrow {b}_{2}$ : solve the second for $\overrightarrow {x}_{2}$ , then substitute into the first and solve for $\overrightarrow {x}_{1}$ .

Write the following system in the form $\mathbf{T}_{n \times n}\overrightarrow {x} = \overrightarrow {b}$ , where $\mathbf{T}$ is block triangular, and then obtain the solution by solving two smaller systems as described above.

\begin{aligned} x_{1} + x_{2} + 3x_{3} + 4x_{4} & = -1, \\ 2x_{3} + 3x_{4} & = 3, \\ x_{1} + 2x_{2} + 5x_{3} + 6x_{4} & = -2, \\ x_{3} + 2x_{4} & = 4. \end{aligned}

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Problem 3.6.11

(a)(b)(c)

Prove that each of the following statements is true for conformable matrices.

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(a)

$\trace (\mathbf{A}\mathbf{B}\mathbf{C}) = \trace (\mathbf{B}\mathbf{C}\mathbf{A}) = \trace (\mathbf{C}\mathbf{A}\mathbf{B})$ .

(b)

$\trace (\mathbf{A}\mathbf{B}\mathbf{C})$ can be different from $\trace (\mathbf{B}\mathbf{A}\mathbf{C})$ .

(c)

$\trace (\mathbf{A}^{T}\mathbf{B}) = \trace (\mathbf{A}\mathbf{B}^{T})$ .

Problem 3.6.12

(a)(b)

Suppose that $\mathbf{A}_{m \times n}$ and $\overrightarrow {x}_{n \times 1}$ have real entries.

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(a)

Prove that $\overrightarrow {x}^{T}\overrightarrow {x} = 0$ if and only if $\overrightarrow {x} = \overrightarrow {0}$ .

(b)

Prove that $\trace (\mathbf{A}^{T}\mathbf{A}) = 0$ if and only if $\mathbf{A} = \mathbf{0}$ .