Math Problems Vault

Problem 8.3.1

(a)(b)(c)

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

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(a)

Show that $A$ is irreducible.

(b)

Find the Perron root and Perron vector for $A$ .

(c)

Find the number of eigenvalues on the spectral circle of $A$ .

Problem 8.3.2

Suppose that the index of imprimitivity of a $5 \times 5$ nonnegative irreducible matrix $A$ is $h = 3$ . Explain why $A$ must be singular with $\text{alg mult}_A(0) = 2$ .

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Problem 8.3.3

Suppose that $A$ is a nonnegative matrix that possesses a positive spectral radius and a corresponding positive eigenvector. Does this force $A$ to be irreducible?

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Problem 8.3.4

Without computing the eigenvalues or the characteristic polynomial, explain why $\sigma (P_n) = \{ 1, \omega , \omega^2, \ldots , \omega^{n-1}\}$ , where $\omega = e^{2\pi i/n}$ for

P_n = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 1 \\ 1 & 0 & 0 & \cdots & 0 \end{pmatrix}.

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Problem 8.3.5

Determine whether

A = \begin{pmatrix} 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & 7 & 0 \\ 2 & 0 & 0 & 0 & 0 \\ 0 & 9 & 2 & 0 & 4 \\ 0 & 0 & 0 & 1 & 0 \end{pmatrix}

is reducible or irreducible.

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Problem 8.3.6

Determine whether the matrix $A$ in Exercise 8.3.5 is primitive or imprimitive.

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Problem 8.3.7

(a)(b)

A matrix $S_{n\times n} \geq 0$ having row sums less than or equal to $1$ with at least one row sum less than $1$ is called a substochastic matrix.

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(a)

Explain why $\rho (S) \leq 1$ for every substochastic matrix.

(b)

Prove that $\rho (S) < 1$ for every irreducible substochastic matrix.

Problem 8.3.8

A nonnegative matrix for which each row sum is $1$ is called a stochastic matrix (some say row-stochastic). Prove that if $A_{n\times n}$ is nonnegative and irreducible with $r = \rho (A)$ , then $A$ is similar to $rP$ for some irreducible stochastic matrix $P$ .

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Remark.

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Consider $D = \begin{pmatrix} p_1 & 0 & \cdots & 0 \\ 0 & p_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & p_n \end{pmatrix}$ , where the $p_k$ 's are the components of the Perron vector for $A$ .

Problem 8.3.9

Wielandt constructed the matrix

W_n = \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & 0 & \cdots & 0 & 1 \\ 1 & 1 & 0 & \cdots & 0 \end{pmatrix}

to show that $W_n^{n^2-2n+2} > 0$ , but $[W_n^{n^2-2n+1}]_{11} = 0$ . Verify that this is true for $n = 4$ .

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Problem 8.3.10

In the Leslie population model, explain what happens to the vector $f(t)$ as $t \to \infty$ depending on whether $r < 1$ , $r = 1$ , or $r > 1$ .

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Remark.

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The Leslie population model divides a population of females into age groups $G_1, \ldots , G_n$ , with $b_k$ the birth rate and $s_k$ the survival rate for group $G_k$ , and $f(t) = (f_1(t), \ldots , f_n(t))^T$ the vector of group sizes at time $t$ , satisfying $f(t+1) = Lf(t)$ for the Leslie matrix $L$ ; $r = \rho (L)$ is the Perron root of $L$ , and $p, q$ are the respective Perron vectors of $L$ and $L^T$ .

Problem 8.3.11

Use the characteristic equation to show that the Leslie matrix (from the Leslie population model) is primitive even if $b_1 = 0$ (assuming all other $b_k$ 's and $s_k$ 's are positive).

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Problem 8.3.12

(a)(b)

A matrix $A \in \mathbb {R}^{n \times n}$ is said to be essentially positive if $A$ is irreducible and $a_{ij} \geq 0$ for every $i \neq j$ . Prove that each of the following statements is equivalent to saying that $A$ is essentially positive.

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(a)

There exists some $\alpha \in \mathbb {R}$ such that $A + \alpha I$ is primitive.

(b)

$e^{tA} > 0$ for all $t > 0$ .

Problem 8.3.13

(a)(b)(c)

Let $A$ be an essentially positive matrix as defined in Exercise 8.3.12. Prove that each of the following statements is true.

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(a)

$A$ has an eigenpair $(\xi , x)$ , where $\xi$ is real and $x > 0$ .

(b)

If $\lambda$ is any eigenvalue for $A$ other than $\xi$ , then $\mathrm{Re}\left(\lambda \right) < \xi$ .

(c)

$\xi$ increases when any entry in $A$ is increased.

Problem 8.3.14

Let $A \geq 0$ be an irreducible matrix, and let $a_{ij}^{(k)}$ denote entries in $A^k$ . Prove that $A$ is primitive if and only if

\rho (A) = \lim _{k\to \infty } \left[a_{ij}^{(k)}\right]^{1/k}.

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