Minimum Polynomials and Krylov Methods

Determine the minimum polynomial for $\mathbf{A} = \begin{pmatrix} 5 & 1 & 2 \\ -4 & 0 & -2 \\ -4 & -1 & -1 \end{pmatrix}$.

Find the minimum polynomial of $\overrightarrow {b} = (-1,1,1)^T$ with respect to the matrix $\mathbf{A}$ given in Exercise 7.11.1.

Use Krylov's method to determine the characteristic polynomial for the matrix $\mathbf{A}$ given in Exercise 7.11.1.

What is the Jordan form for a matrix whose minimum polynomial is $m(x) = (x-\lambda )(x-\mu )^2$ and whose characteristic polynomial is $c(x) = (x-\lambda )^2(x-\mu )^4$?

Use the technique for computing a matrix's minimum polynomial via an orthogonalized Krylov sequence (successively Gram--Schmidt-orthogonalizing $\mathbf{I}, \mathbf{A}, \mathbf{A}^2, \ldots$ in the Frobenius inner product $\langle \mathbf{X} \mid \mathbf{Y} \rangle = \operatorname {trace}(\mathbf{X}^T\mathbf{Y})$ until a power $\mathbf{A}^k$ is found to be a linear combination of the earlier orthogonalized terms $\mathbf{U}_0, \ldots , \mathbf{U}_{k-1}$, with coefficients recovered by solving $\mathbf{R}\overrightarrow {c} = \mathbf{R}^{-1}$ triangular system for the $\alpha_i$'s) to determine the minimum polynomial for

Explain why similar matrices have the same minimum and characteristic polynomials.

Show that two matrices can have the same minimum and characteristic polynomials without being similar by considering $\mathbf{A} = \begin{pmatrix} \mathbf{N} & \mathbf{0} \\ \mathbf{0} & \mathbf{N} \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} \mathbf{N} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} \end{pmatrix}$, where $\mathbf{N} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.

Prove that if $\mathbf{A}$ and $\mathbf{B}$ are nonderogatory matrices that have the same characteristic polynomial, then $\mathbf{A}$ is similar to $\mathbf{B}$.

Use the Lanczos algorithm to find an orthogonal matrix $\mathbf{P}$ such that $\mathbf{P}^T\mathbf{A}\mathbf{P} = \mathbf{T}$ is tridiagonal, where $\mathbf{A} = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$.

Starting with $\overrightarrow {x}_0 = \overrightarrow {0}$, apply the conjugate gradient algorithm to solve $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$, where $\mathbf{A} = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ and $\overrightarrow {b} = \begin{pmatrix} 4 \\ 0 \\ 0 \end{pmatrix}$.

Use Arnoldi's algorithm to find an orthogonal matrix $\mathbf{Q}$ such that $\mathbf{Q}^T\mathbf{A}\mathbf{Q} = \mathbf{H}$ is upper Hessenberg, where $\mathbf{A} = \begin{pmatrix} 5 & 1 & 2 \\ -4 & 0 & -2 \\ -4 & -1 & -1 \end{pmatrix}$.

Use GMRES to solve $\mathbf{A}\overrightarrow {x} = \overrightarrow {b}$ for $\mathbf{A} = \begin{pmatrix} 5 & 1 & 2 \\ -4 & 0 & -2 \\ -4 & -1 & -1 \end{pmatrix}$ and $\overrightarrow {b} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$.