Functions of Diagonalizable Matrices

Determine $\cos \mathbf{A}$ for $\mathbf{A} = \begin{pmatrix} -\pi /2 & \pi /2 \\ \pi /2 & -\pi /2 \end{pmatrix}$.

For the matrix $\mathbf{A}$ in Example 7.3.3, verify with direct computation that $\mathrm{e}^{\lambda_1 t}\mathbf{G}_1 + \mathrm{e}^{\lambda_2 t}\mathbf{G}_2 = \mathbf{P}\begin{pmatrix} \mathrm{e}^{\lambda_1 t} & 0 \\ 0 & \mathrm{e}^{\lambda_2 t} \end{pmatrix}\mathbf{P}^{-1} = \mathrm{e}^{\mathbf{A}t}$.

Example 7.3.3's matrix is $\mathbf{A} = \begin{pmatrix} 0 & \alpha \\ \beta & -(\alpha +\beta ) \end{pmatrix}$, for constants $\alpha , \beta > 0$.

Explain why $\sin^2 \mathbf{A} + \cos^2 \mathbf{A} = \mathbf{I}$ for a diagonalizable matrix $\mathbf{A}$.

Explain why $\mathrm{e}^{\mathbf{0}} = \mathbf{I}$ for every square zero matrix.

The spectral mapping property for diagonalizable matrices says that if $f(\mathbf{A})$ exists, and if $\left\{ \lambda_1,\lambda_2,\ldots ,\lambda_n\right\}$ are the eigenvalues of $\mathbf{A}_{n\times n}$ (including multiplicities), then $\left\{ f(\lambda_1),\ldots ,f(\lambda_n)\right\}$ are the eigenvalues of $f(\mathbf{A})$.

Explain why $\operatorname {det}\left(\mathrm{e}^{\mathbf{A}}\right) = \mathrm{e}^{\operatorname {trace}(\mathbf{A})}$.

Suppose that for nondiagonalizable matrices $\mathbf{A}_{m\times m}$ an infinite series $f(z) = \sum_{n=0}^\infty c_n(z-z_0)^n$ is used to define $f(\mathbf{A}) = \sum_{n=0}^\infty c_n(\mathbf{A}-z_0\mathbf{I})^n$. Neglecting convergence issues, explain why there is a polynomial $p(z)$ of at most degree $m-1$ such that $f(\mathbf{A}) = p(\mathbf{A})$.

If $f(\mathbf{A})$ exists for a diagonalizable $\mathbf{A}$, explain why $\mathbf{A}f(\mathbf{A}) = f(\mathbf{A})\mathbf{A}$. What can you say when $\mathbf{A}$ is not diagonalizable?

Explain why $\mathrm{e}^{\mathbf{A}+\mathbf{B}} = \mathrm{e}^{\mathbf{A}}\mathrm{e}^{\mathbf{B}}$ whenever $\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}$. Give an example to show that $\mathrm{e}^{\mathbf{A}+\mathbf{B}}$, $\mathrm{e}^{\mathbf{A}}\mathrm{e}^{\mathbf{B}}$, and $\mathrm{e}^{\mathbf{B}}\mathrm{e}^{\mathbf{A}}$ all can differ when $\mathbf{A}\mathbf{B} \ne \mathbf{B}\mathbf{A}$.

Hint: Exercise 7.2.16 can be used for the diagonalizable case. For the general case, consider $\mathbf{F}(t) = \mathrm{e}^{(\mathbf{A}+\mathbf{B})t} - \mathrm{e}^{\mathbf{A}t}\mathrm{e}^{\mathbf{B}t}$ and $\mathbf{F}'(t)$.

Show that $\mathrm{e}^{\mathbf{A}}$ is an orthogonal matrix whenever $\mathbf{A}$ is skew symmetric.

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 the OFF state.

The spectral radius of $\mathbf{A}$ is $\rho (\mathbf{A}) = \max_{\lambda_i \in \sigma (\mathbf{A})} \left|\lambda_i\right|$. Prove that if $\mathbf{A}$ is diagonalizable, then

Note: This result is true for nondiagonalizable matrices as well, but the proof at this point is more involved.

Find a dominant eigenpair for $\mathbf{A} = \begin{pmatrix} 7 & 2 & 3 \\ 0 & 2 & 0 \\ -6 & -2 & -2 \end{pmatrix}$ by the power method.

Apply the inverse power method to find an eigenvector for each of the eigenvalues of the matrix $\mathbf{A}$ in Exercise 7.3.13.

Explain why the function $m(\overrightarrow {v})$ used in the development of the power method is not a continuous function, so statements like $m(\overrightarrow {x}_n) \to m(\overrightarrow {x})$ when $\overrightarrow {x}_n \to \overrightarrow {x}$ are not valid. Nevertheless, if $\lim_{n\to \infty } \overrightarrow {x}_n \ne \overrightarrow {0}$, then $\lim_{n\to \infty } m(\overrightarrow {x}_n) \ne 0$.

Let $\mathbf{H} = \begin{pmatrix} 1 & 0 & 0 \\ -1 & -2 & -1 \\ 0 & 2 & 1 \end{pmatrix}$.

Show that the QR iteration can fail to converge using $\mathbf{H} = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$.