Krylov subspace methods

The entries of the matrix $\left\| b_{\mu \nu }\right\|$ are determined by the formulas:

where $b_{11}, b_{12}, \ldots , b_{1n}$ and $a_{\mu \nu }$ are given numbers. Find the value of the product of the determinants

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

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

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 $\overrightarrow {b} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}$.