Math Problems Vault

Problem 7.1.1

Determine the eigenvalues and eigenvectors for the following matrices.

\mathbf{A} = \begin{pmatrix} -10 & -7 \\ 14 & 11 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 16 & 8 \\ 4 & 14 & 8 \\ -8 & -32 & -18 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} 3 & -2 & 5 \\ 0 & 1 & 4 \\ 0 & -1 & 5 \end{pmatrix},

\mathbf{D} = \begin{pmatrix} 0 & 6 & 3 \\ -1 & 5 & 1 \\ -1 & 2 & 4 \end{pmatrix}, \quad \mathbf{E} = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix}.

Which, if any, are deficient in eigenvectors in the sense that there fails to exist a complete linearly independent set?

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Problem 7.1.2

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

Without doing an eigenvalue--eigenvector computation, determine which of the following are eigenvectors for

\mathbf{A} = \begin{pmatrix} -9 & -6 & -2 & -4 \\ -8 & -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32 & 21 & 7 & 12 \end{pmatrix},

and for those which are eigenvectors, identify the associated eigenvalue.

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(a)

$(-1,1,0,1)^T$ .

(b)

$(1,0,-1,0)^T$ .

(c)

$(-1,0,2,2)^T$ .

(d)

$(0,1,-3,0)^T$ .

Problem 7.1.3

Explain why the eigenvalues of triangular and diagonal matrices

\mathbf{T} = \begin{pmatrix} t_{11} & t_{12} & \cdots & t_{1n} \\ 0 & t_{22} & \cdots & t_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & t_{nn} \end{pmatrix} \quad \text{and} \quad \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}

are simply the diagonal entries---the $t_{ii}$ 's and $\lambda_i$ 's.

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Problem 7.1.4

For $\mathbf{T} = \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{0} & \mathbf{C} \end{pmatrix}$ , prove $\operatorname {det}\left(\mathbf{T}-\lambda \mathbf{I}\right) = \operatorname {det}\left(\mathbf{A}-\lambda \mathbf{I}\right)\operatorname {det}\left(\mathbf{C}-\lambda \mathbf{I}\right)$ to conclude that $\sigma \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{0} & \mathbf{C} \end{pmatrix} = \sigma (\mathbf{A}) \cup \sigma (\mathbf{C})$ for square $\mathbf{A}$ and $\mathbf{C}$ .

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Problem 7.1.5

Determine the eigenvectors of $\mathbf{D} = \operatorname {diag}(\lambda_1, \lambda_2, \ldots , \lambda_n)$ . In particular, what is the eigenspace associated with $\lambda_i$ ?

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Problem 7.1.6

Prove that $0 \in \sigma (\mathbf{A})$ if and only if $\mathbf{A}$ is a singular matrix.

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Problem 7.1.7

Explain why it's apparent that

\mathbf{A}_{n\times n} = \begin{pmatrix} n & 1 & 1 & \cdots & 1 \\ 1 & n & 1 & \cdots & 1 \\ 1 & 1 & n & \cdots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & n \end{pmatrix}

doesn't have a zero eigenvalue, and hence why $\mathbf{A}$ is nonsingular.

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Problem 7.1.8

Explain why the eigenvalues of $\mathbf{A}^{*}\mathbf{A}$ and $\mathbf{A}\mathbf{A}^{*}$ are real and nonnegative for every $\mathbf{A} \in \mathcal{C}^{m\times n}$ .

Hint: Consider $\left\| \mathbf{A}\overrightarrow {x}\right\|_2^2 / \left\| \overrightarrow {x}\right\|_2^2$ . When are the eigenvalues of $\mathbf{A}^{*}\mathbf{A}$ and $\mathbf{A}\mathbf{A}^{*}$ strictly positive?

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Problem 7.1.9

(a)(b)

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(a)

If $\mathbf{A}$ is nonsingular, and if $(\lambda , \overrightarrow {x})$ is an eigenpair for $\mathbf{A}$ , show that $(\lambda^{-1}, \overrightarrow {x})$ is an eigenpair for $\mathbf{A}^{-1}$ .

(b)

For all $\alpha \notin \sigma (\mathbf{A})$ , prove that $\overrightarrow {x}$ is an eigenvector of $\mathbf{A}$ if and only if $\overrightarrow {x}$ is an eigenvector of $(\mathbf{A}-\alpha \mathbf{I})^{-1}$ .

Problem 7.1.10

(a)(b)

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(a)

Show that if $(\lambda , \overrightarrow {x})$ is an eigenpair for $\mathbf{A}$ , then $(\lambda^k, \overrightarrow {x})$ is an eigenpair for $\mathbf{A}^k$ for each positive integer $k$ .

(b)

If $p(x) = \alpha_0 + \alpha_1 x + \alpha_2 x^2 + \cdots + \alpha_k x^k$ is any polynomial, then we define $p(\mathbf{A})$ to be the matrix

p(\mathbf{A}) = \alpha _0 \mathbf{I} + \alpha _1 \mathbf{A} + \alpha _2 \mathbf{A}^2 + \cdots + \alpha _k \mathbf{A}^k.

Show that if $(\lambda , \overrightarrow {x})$ is an eigenpair for $\mathbf{A}$ , then $(p(\lambda ), \overrightarrow {x})$ is an eigenpair for $p(\mathbf{A})$ .

Problem 7.1.11

Explain why Gerschgorin's theorem implies that

\mathbf{A} = \begin{pmatrix} 1 & 0 & -2 & 0 \\ 0 & 12 & 0 & -4 \\ 1 & 0 & -1 & 0 \\ 0 & 5 & 0 & 0 \end{pmatrix}

must have at least two real eigenvalues. Corroborate this fact by computing the eigenvalues of $\mathbf{A}$ .

Gerschgorin's theorem: if the $n$ Gerschgorin circles $G_i = \left\{ z \in \mathcal{C} \mid \left|z - a_{ii}\right| \le \sum_{j \ne i} \left|a_{ij}\right|\right\}$ associated with the rows of an $n \times n$ matrix $\mathbf{A}$ are such that $k$ of them form a connected region that is disjoint from the remaining $n-k$ circles, then exactly $k$ of the eigenvalues of $\mathbf{A}$ lie in the connected region formed by the $k$ circles, and the remaining $n-k$ eigenvalues lie in the region formed by the union of the other $n-k$ circles.

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Problem 7.1.12

If $\mathbf{A}$ is nilpotent ( $\mathbf{A}^k = \mathbf{0}$ for some $k$ ), explain why $\operatorname {trace}(\mathbf{A}) = 0$ .

Hint: What is $\sigma (\mathbf{A})$ ?

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Problem 7.1.13

If $\overrightarrow {x}_1, \overrightarrow {x}_2, \ldots , \overrightarrow {x}_k$ are eigenvectors of $\mathbf{A}$ associated with the same eigenvalue $\lambda$ , explain why every nonzero linear combination

\overrightarrow {v} = \alpha _1 \overrightarrow {x}_1 + \alpha _2 \overrightarrow {x}_2 + \cdots + \alpha _n \overrightarrow {x}_n

is also an eigenvector for $\mathbf{A}$ associated with the eigenvalue $\lambda$ .

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Problem 7.1.14

Explain why an eigenvector for a square matrix $\mathbf{A}$ cannot be associated with two distinct eigenvalues for $\mathbf{A}$ .

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Problem 7.1.15

Suppose $\sigma (\mathbf{A}_{n\times n}) = \sigma (\mathbf{B}_{n\times n})$ . Does this guarantee that $\mathbf{A}$ and $\mathbf{B}$ have the same characteristic polynomial?

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Problem 7.1.16

(a)(b)

Construct $2 \times 2$ examples to prove the following statements.

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(a)

$\lambda \in \sigma (\mathbf{A})$ and $\mu \in \sigma (\mathbf{B}) \centernot \implies \lambda +\mu \in \sigma (\mathbf{A}+\mathbf{B})$ .

(b)

$\lambda \in \sigma (\mathbf{A})$ and $\mu \in \sigma (\mathbf{B}) \centernot \implies \lambda \mu \in \sigma (\mathbf{A}\mathbf{B})$ .

Problem 7.1.17

(a)(b)(c)

Suppose that $\left\{ \lambda_1, \lambda_2, \ldots , \lambda_n\right\}$ are the eigenvalues for $\mathbf{A}_{n\times n}$ , and let $(\lambda_k, \overrightarrow {c})$ be a particular eigenpair.

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(a)

For $\lambda \notin \sigma (\mathbf{A})$ , explain why $(\mathbf{A}-\lambda \mathbf{I})^{-1}\overrightarrow {c} = \overrightarrow {c}/(\lambda_k-\lambda )$ .

(b)

For an arbitrary vector $\overrightarrow {d}_{n\times 1}$ , prove that the eigenvalues of $\mathbf{A}+\overrightarrow {c}\overrightarrow {d}^T$ agree with those of $\mathbf{A}$ except that $\lambda_k$ is replaced by $\lambda_k + \overrightarrow {d}^T\overrightarrow {c}$ .

(c)

How can $\overrightarrow {d}$ be selected to guarantee that the eigenvalues of $\mathbf{A}+\overrightarrow {c}\overrightarrow {d}^T$ and $\mathbf{A}$ agree except that $\lambda_k$ is replaced by a specified number $\mu$ ?

Problem 7.1.18

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

Suppose that $\mathbf{A}$ is a square matrix.

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(a)

Explain why $\mathbf{A}$ and $\mathbf{A}^T$ have the same eigenvalues.

(b)

Explain why $\lambda \in \sigma (\mathbf{A}) \iff \overline{\lambda } \in \sigma (\mathbf{A}^{*})$ .

Hint: Recall Exercise 6.1.8.

(c)

Do these results imply that $\lambda \in \sigma (\mathbf{A}) \iff \overline{\lambda } \in \sigma (\mathbf{A})$ when $\mathbf{A}$ is a square matrix of real numbers?

(d)

A nonzero row vector $\overrightarrow {y}^{*}$ is called a left-hand eigenvector for $\mathbf{A}$ whenever there is a scalar $\mu \in \mathcal{C}$ such that $\overrightarrow {y}^{*}(\mathbf{A}-\mu \mathbf{I}) = \overrightarrow {0}$ . Explain why $\mu$ must be an eigenvalue for $\mathbf{A}$ in the ``right-hand'' sense of the term when $\mathbf{A}$ is a square matrix of real numbers.

Problem 7.1.19

(a)(b)

Consider matrices $\mathbf{A}_{m\times n}$ and $\mathbf{B}_{n\times m}$ .

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(a)

Explain why $\mathbf{A}\mathbf{B}$ and $\mathbf{B}\mathbf{A}$ have the same characteristic polynomial if $m=n$ .

Hint: Recall Exercise 6.2.16.

(b)

Explain why the characteristic polynomials for $\mathbf{A}\mathbf{B}$ and $\mathbf{B}\mathbf{A}$ can't be the same when $m \ne n$ , and then explain why $\sigma (\mathbf{A}\mathbf{B})$ and $\sigma (\mathbf{B}\mathbf{A})$ agree, with the possible exception of a zero eigenvalue.

Problem 7.1.20

If $\mathbf{A}\mathbf{B} = \mathbf{B}\mathbf{A}$ , prove that $\mathbf{A}$ and $\mathbf{B}$ have a common eigenvector.

Hint: For $\lambda \in \sigma (\mathbf{A})$ , let the columns of $\mathbf{X}$ be a basis for $N(\mathbf{A}-\lambda \mathbf{I})$ so that $(\mathbf{A}-\lambda \mathbf{I})\mathbf{B}\mathbf{X} = \mathbf{0}$ . Explain why there exists a matrix $\mathbf{P}$ such that $\mathbf{B}\mathbf{X} = \mathbf{X}\mathbf{P}$ , and then consider any eigenpair for $\mathbf{P}$ .

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Problem 7.1.21

(a)(b)

For fixed matrices $\mathbf{P}_{m\times m}$ and $\mathbf{Q}_{n\times n}$ , let $\mathbf{T}$ be the linear operator on $\mathcal{C}^{m\times n}$ defined by $\mathbf{T}(\mathbf{A}) = \mathbf{P}\mathbf{A}\mathbf{Q}$ .

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(a)

Show that if $\overrightarrow {x}$ is a right-hand eigenvector for $\mathbf{P}$ and $\overrightarrow {y}^{*}$ is a left-hand eigenvector for $\mathbf{Q}$ , then $\overrightarrow {x}\overrightarrow {y}^{*}$ is an eigenvector for $\mathbf{T}$ .

(b)

Explain why $\operatorname {trace}(\mathbf{T}) = \operatorname {trace}(\mathbf{P})\operatorname {trace}(\mathbf{Q})$ .

Problem 7.1.22

(a)(b)

Let $\mathbf{D} = \operatorname {diag}(\lambda_1, \lambda_2, \ldots , \lambda_n)$ be a diagonal real matrix such that $\lambda_1 < \lambda_2 < \cdots < \lambda_n$ , and let $\overrightarrow {v}_{n\times 1}$ be a column of real nonzero numbers.

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(a)

Prove that if $\alpha$ is real and nonzero, then $\lambda_i$ is not an eigenvalue for $\mathbf{D}+\alpha \overrightarrow {v}\overrightarrow {v}^T$ . Show that the eigenvalues of $\mathbf{D}+\alpha \overrightarrow {v}\overrightarrow {v}^T$ are in fact given by the solutions of the secular equation $f(\xi )=0$ defined by

f(\xi ) = 1 + \alpha \sum _{i=1}^n \frac{v_i^2}{\lambda _i - \xi }.

For $n=4$ and $\alpha >0$ , verify that the graph of $f(\xi )$ has $n$ vertical asymptotes at the $\lambda_i$ 's, with $f$ increasing from $0$ to $+\infty$ on $(\lambda_i, \lambda_{i+1})$ and a further branch beyond $\lambda_n$ approaching the horizontal asymptote at $1$ , so that each interval $(\lambda_i, \lambda_{i+1})$ and the ray $(\lambda_n, \infty )$ contains exactly one root $\xi_i$ of $f$ , and thereby conclude that the eigenvalues of $\mathbf{D}+\alpha \overrightarrow {v}\overrightarrow {v}^T$ interlace with those of $\mathbf{D}$ .

Figure 7.1.4

(b)

Verify that $(\mathbf{D}-\xi_i\mathbf{I})^{-1}\overrightarrow {v}$ is an eigenvector for $\mathbf{D}+\alpha \overrightarrow {v}\overrightarrow {v}^T$ that is associated with the eigenvalue $\xi_i$ .

Problem 7.1.23

(a)(b)(c)

Let $\lambda_1, \ldots , \lambda_n$ be the roots of the polynomial $p(\lambda ) = \lambda^n + c_1\lambda^{n-1} + c_2\lambda^{n-2} + \cdots + c_n$ , and let $\tau_k = \lambda_1^k + \lambda_2^k + \cdots + \lambda_n^k$ . Newton's identities say $c_k = -(\tau_1 c_{k-1} + \tau_2 c_{k-2} + \cdots + \tau_{k-1}c_1 + \tau_k)/k$ . Derive these identities by executing the following steps:

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(a)

Show $p'(\lambda ) = p(\lambda )\sum_{i=1}^n (\lambda -\lambda_i)^{-1}$ (logarithmic differentiation).

(b)

Use the geometric series expansion for $(\lambda -\lambda_i)^{-1}$ to show that for $\left|\lambda \right| > \max_i \left|\lambda_i\right|$ ,

\sum _{i=1}^n \frac{1}{\lambda -\lambda _i} = \frac{n}{\lambda } + \frac{\tau _1}{\lambda ^2} + \frac{\tau _2}{\lambda ^3} + \cdots .

(c)

Combine these two results, and equate like powers of $\lambda$ .

Problem 7.1.24

Let the characteristic equation for $\mathbf{A}$ be given by $\lambda^n + c_1\lambda^{n-1} + c_2\lambda^{n-2} + \cdots + c_n = 0$ , and define a sequence by taking $\mathbf{B}_0 = \mathbf{I}$ and

\mathbf{B}_k = -\frac{\operatorname {trace}(\mathbf{A}\mathbf{B}_{k-1})}{k}\mathbf{I} + \mathbf{A}\mathbf{B}_{k-1} \quad \text{for } k = 1, 2, \ldots , n.

Prove that for each $k$ ,

c_k = -\frac{\operatorname {trace}(\mathbf{A}\mathbf{B}_{k-1})}{k}.

Hint: Use Newton's identities (Exercise 7.1.23), and recall Exercise 7.1.10(a).

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