Math Problems Vault

Problem 4.8.1

Explain why rank is a similarity invariant.

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Problem 4.8.2

Explain why similarity is transitive in the sense that $\mathbf{A} \simeq \mathbf{B}$ and $\mathbf{B} \simeq \mathbf{C}$ implies $\mathbf{A} \simeq \mathbf{C}$ .

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Problem 4.8.3

(a)(b)

$\mathbf{A}(x,y,z) = (x+2y-z, \; -y, \; x+7z)$ is a linear operator on $\mathbb {R}^{3}$ .

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(a)

Determine $[\mathbf{A}]_{\mathcal{S}}$ , where $\mathcal{S}$ is the standard basis.

(b)

Determine $[\mathbf{A}]_{\mathcal{S}'}$ as well as the nonsingular matrix $\mathbf{Q}$ such that $[\mathbf{A}]_{\mathcal{S}'} = \mathbf{Q}^{-1}[\mathbf{A}]_{\mathcal{S}}\mathbf{Q}$ for $\mathcal{S}' = \left\{ \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\right\}$ .

Problem 4.8.4

Let $\mathbf{A} = \begin{pmatrix} 1 & 2 & 0 \\ 3 & 1 & 4 \\ 0 & 1 & 5 \end{pmatrix}$ and $\mathcal{B} = \left\{ \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}\right\}$ . Consider $\mathbf{A}$ as a linear operator on $\mathbb {R}^{n \times 1}$ by means of matrix multiplication $\mathbf{A}(\overrightarrow {x}) = \mathbf{A}\overrightarrow {x}$ , and determine $[\mathbf{A}]_{\mathcal{B}}$ .

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Problem 4.8.5

Show that $\mathbf{C} = \begin{pmatrix} 4 & 6 \\ 3 & 4 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} -2 & -3 \\ 6 & 10 \end{pmatrix}$ are similar matrices, and find a nonsingular matrix $\mathbf{Q}$ such that $\mathbf{C} = \mathbf{Q}^{-1}\mathbf{B}\mathbf{Q}$ .

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Problem 4.8.6

Let $\mathbf{T}$ be the linear operator $\mathbf{T}(x,y) = (-7x-15y, \; 6x+12y)$ . Find a basis $\mathcal{B}$ such that $[\mathbf{T}]_{\mathcal{B}} = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ , and determine a matrix $\mathbf{Q}$ such that $[\mathbf{T}]_{\mathcal{B}} = \mathbf{Q}^{-1}[\mathbf{T}]_{\mathcal{S}}\mathbf{Q}$ , where $\mathcal{S}$ is the standard basis.

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Problem 4.8.7

By considering the rotator $\mathbf{Q}(x,y) = (x\cos \theta - y\sin \theta , \; x\sin \theta + y\cos \theta )$ , show that the matrices

\mathbf{R} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \quad \text{and} \quad \mathbf{D} = \begin{pmatrix} e^{i\theta } & 0 \\ 0 & e^{-i\theta } \end{pmatrix}

are similar over the complex field.

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Problem 4.8.8

(a)(b)

Let $\lambda$ be a scalar such that $(\mathbf{C} - \lambda \mathbf{I})_{n \times n}$ is singular.

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(a)

If $\mathbf{B} \simeq \mathbf{C}$ , prove that $(\mathbf{B} - \lambda \mathbf{I})$ is also singular.

(b)

Prove that $(\mathbf{B} - \lambda_i \mathbf{I})$ is singular whenever $\mathbf{B}_{n \times n}$ is similar to

\mathbf{D} = \begin{pmatrix} \lambda _1 & 0 & \cdots & 0 \\ 0 & \lambda _2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda _n \end{pmatrix}.

Problem 4.8.9

If $\mathbf{A} \simeq \mathbf{B}$ , show that $\mathbf{A}^k \simeq \mathbf{B}^k$ for all nonnegative integers $k$ .

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Problem 4.8.10

(a)(b)

Suppose $\mathcal{B} = \left\{ \overrightarrow {x}_1, \overrightarrow {x}_2, \ldots , \overrightarrow {x}_n\right\}$ and $\mathcal{B}' = \left\{ \overrightarrow {y}_1, \overrightarrow {y}_2, \ldots , \overrightarrow {y}_n\right\}$ are bases for an $n$ -dimensional subspace $\mathcal{V} \subseteq \mathbb {R}^{m \times 1}$ , and let $\mathbf{X}_{m \times n}$ and $\mathbf{Y}_{m \times n}$ be the matrices whose columns are the vectors from $\mathcal{B}$ and $\mathcal{B}'$ , respectively.

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(a)

Explain why $\mathbf{Y}^T \mathbf{Y}$ is nonsingular, and prove that the change of basis matrix from $\mathcal{B}$ to $\mathcal{B}'$ is $\mathbf{P} = \left(\mathbf{Y}^T \mathbf{Y}\right)^{-1} \mathbf{Y}^T \mathbf{X}$ .

(b)

Describe $\mathbf{P}$ when $m = n$ .

Problem 4.8.11

(a)(b)(c)

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(a)

$\mathbf{N}$ is nilpotent of index $k$ when $\mathbf{N}^k = \mathbf{0}$ but $\mathbf{N}^{k-1} \neq \mathbf{0}$ . If $\mathbf{N}$ is a nilpotent operator of index $n$ on $\mathbb {R}^{n}$ , and if $\mathbf{N}^{n-1}(\overrightarrow {y}) \neq \overrightarrow {0}$ , show $\mathcal{B} = \left\{ \overrightarrow {y}, \mathbf{N}(\overrightarrow {y}), \mathbf{N}^2(\overrightarrow {y}), \ldots , \mathbf{N}^{n-1}(\overrightarrow {y})\right\}$ is a basis for $\mathbb {R}^{n}$ , and then demonstrate that

[\mathbf{N}]_{\mathcal{B}} = \mathbf{J} = \begin{pmatrix} 0 & 0 & \cdots & 0 & 0 \\ 1 & 0 & \cdots & 0 & 0 \\ 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 \end{pmatrix}.

(b)

If $\mathbf{A}$ and $\mathbf{B}$ are any two $n \times n$ nilpotent matrices of index $n$ , explain why $\mathbf{A} \simeq \mathbf{B}$ .

(c)

Explain why all $n \times n$ nilpotent matrices of index $n$ must have a zero trace and be of rank $n-1$ .

Problem 4.8.12

(a)(b)(c)(d)

$\mathbf{E}$ is idempotent when $\mathbf{E}^2 = \mathbf{E}$ . For an idempotent operator $\mathbf{E}$ on $\mathbb {R}^{n}$ , let $\mathcal{X} = \left\{ \overrightarrow {x}_i\right\}_{i=1}^r$ and $\mathcal{Y} = \left\{ \overrightarrow {y}_i\right\}_{i=1}^{n-r}$ be bases for $R(\mathbf{E})$ and $N(\mathbf{E})$ , respectively.

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(a)

Prove that $\mathcal{B} = \mathcal{X} \cup \mathcal{Y}$ is a basis for $\mathbb {R}^{n}$ .

(b)

Show that $[\mathbf{E}]_{\mathcal{B}} = \begin{pmatrix} \mathbf{I}_r & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{pmatrix}$ .

(c)

Explain why two $n \times n$ idempotent matrices of the same rank must be similar.

(d)

If $\mathbf{F}$ is an idempotent matrix, prove that $\operatorname {rk}\left(\mathbf{F}\right) = \operatorname {trace}(\mathbf{F})$ .