Positive Matrices

Verify Perron's theorem by computing the eigenvalues and eigenvectors for

Find the right-hand Perron vector $p$ as well as the left-hand Perron vector $q^T$.

Convince yourself that (8.2.2)-(8.2.5) are indeed true.

Statements (8.2.2)-(8.2.5) are properties of the Perron root and Perron vector established in the text for a positive matrix $A_{n\times n} > 0$ with $r = \rho (A)$: (8.2.2) $r > 0$; (8.2.3) $r \in \sigma (A)$ (called the Perron root); (8.2.4) $\text{alg mult}_A(r) = 1$; (8.2.5) there exists an eigenvector $x > 0$ such that $Ax = rx$.

Provide the details that explain why the Perron vector is uniquely defined.

Find the Perron root and the Perron vector for

where $\alpha + \beta = 1$ with $\alpha , \beta > 0$.

Suppose that $A_{n\times n} > 0$ has $\rho (A) = r$.

Prove that if every row (or column) sum of $A_{n\times n} > 0$ is equal to $\rho$, then $\rho (A) = \rho$.

Prove that if $A_{n\times n} > 0$, then

To show the extent to which the hypothesis of positivity cannot be relaxed in Perron's theorem, construct examples of square matrices $A$ such that $A \geq 0$, but $A \not> 0$ (i.e., $A$ has at least one zero entry), with $r = \rho (A) \in \sigma (A)$ that demonstrate the validity of the following statements. Different examples may be used for the different statements.

Establish the min-max version of the Collatz--Wielandt formula that says the Perron root for $A > 0$ is given by $r = \min_{x \in \mathcal{P}} g(x)$, where

Notice that $\mathcal{N} = \left\{ x \mid x \geq 0 \text{ with } x \neq 0\right\}$ is used in the max-min version of the Collatz--Wielandt formula, but $\mathcal{P} = \left\{ x \mid x > 0\right\}$ is used in the min-max version in the previous exercise. Give an example of a matrix $A > 0$ that shows $r \neq \min_{x \in \mathcal{N}} g(x)$ when $g(x)$ is defined as