Math Problems Vault

Problem 1201

Find the $k$ -th power $A^{k}$ of the Jordan block

A =\left(\begin{smallmatrix} \alpha & 1 & 0 & 0 & \ldots & 0 & 0 \\ 0 & \alpha & 1 & 0 & \ldots & 0 & 0 \\ 0 & 0 & \alpha & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & \alpha & 1 \\ 0 & 0 & 0 & 0 & \ldots & 0 & \alpha \end{smallmatrix}\right) \text{ of order } n

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Problem 1202

Prove that the value of the polynomial $f(x)$ of the Jordan block (11.1) is given by the formula

f(A) =\left(\begin{smallmatrix} f(\alpha ) & \frac{f^{\prime }(\alpha )}{1 !} & \frac{f^{\prime \prime }(\alpha )}{2 !} & \frac{f^{\prime \prime \prime }(\alpha )}{3 !} & \ldots & \frac{f^{(n-1)}(\alpha )}{(n-1) !} \\ 0 & f(\alpha ) & \frac{f^{\prime }(\alpha )}{1 !} & \frac{f^{\prime \prime }(\alpha )}{2 !} & \ldots & \frac{f^{(n-2)}(\alpha )}{(n-2) !} \\ \ldots \ldots & \ldots & \ldots \ldots \ldots \ldots \ldots \\ 0 & 0 & 0 & 0 & \ldots & f(\alpha ) \end{smallmatrix}\right)

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Problem 1203

Let $\mu_{\mathrm{A}}(\lambda ) = \left(\lambda -\lambda_{1}\right)^{r_{1}} \ldots \left(\lambda -\lambda_{s}\right)^{r_{s}}$ be the minimal polynomial of the operator A. If for each $k = 1, \ldots , s$ the function $f(\lambda )$ is defined at the point $\lambda_{k}$ and is differentiable there at least $r_{k}-1$ times, then $f(\lambda )$ is said to be defined on the spectrum of the operator A, and the system of numbers

\left(f^{(j)}\left(\lambda _{k}\right)\right)_{1 \leqslant k \leqslant s ; 1 \leqslant j \leqslant r_{k}-1}

is called the system of values of the function $f(\lambda )$ on the spectrum of the operator A. Prove that the values of the polynomials $g(\lambda )$ and $h(\lambda )$ of the operator A coincide if and only if the values of these polynomials on the spectrum of the operator $\mathbf{A}$ coincide.

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Problem 1204

(1)(2)

Let the function $f(\lambda )$ be defined on the spectrum of the operator $\mathbf{A}$ with minimal polynomial $\mu_{\mathrm{A}}(\lambda ) = \left(\lambda -\lambda_{1}\right)^{r_{1}} \ldots \left(\lambda -\lambda_{s}\right)^{r_{s}}$ . Consider the polynomial

r(\lambda ) = \sum _{k = 1}^{s}\left(\alpha _{k, 1}+\alpha _{k, 2}\left(\lambda -\lambda _{k}\right)+\ldots +\alpha _{k, r_{s}}\left(\lambda -\lambda _{k}\right)^{r_{k}-1}\right) \cdot \frac{\mu _{\mathrm{A}}(\lambda )}{\left(\lambda -\lambda _{k}\right)^{r_{k}}}

where

\alpha _{k, j} = \frac{1}{(j-1) !}\left(\frac{\left(\lambda -\lambda _{k}\right)^{r_{k}} f(\lambda )}{\mu _{\mathrm{A}}(\lambda )}\right)_{\lambda = \lambda _{k}}^{(j-1)}

$j = 1,2, \ldots , r_{k}, k = 1,2, \ldots , s$ . Prove that:

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(1)

the systems of values of the polynomial $r(\lambda )$ and of the function $f(\lambda )$ on the spectrum of the operator A coincide;

(2)

among all polynomials with this property, $r(\lambda )$ has the smallest degree.

This polynomial $r(\lambda )$ is called the interpolation polynomial of the function $f(\lambda )$ on the spectrum of the operator A. The value $f(\mathbf{A})$ of the function $f$ of the operator A is by definition set equal to $r(\mathbf{A})$ .

Problem 1205

Compute $A^{100}$ , where

A =\left(\begin{smallmatrix} 10 & 1 & 5 \\ 13 & 2 & 7 \\ -20 & -2 & -10 \end{smallmatrix}\right)

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Problem 1206

Compute $A^{101}$ , where

A =\left(\begin{smallmatrix} -2 & -3 & -4 \\ -1 & 0 & 0 \\ 1 & 1 & 1 \end{smallmatrix}\right)

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Problem 1207

(1)(2)(3)

Compute $\exp A$ , where:

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(1)

$A = \left(\begin{array}{rr}0 & -3 \\ 3 & 0\end{array}\right)$ ;

(2)

$A = \left(\begin{array}{rrr}0 & -1 & -1 \\ -1 & -1 & -2 \\ 0 & 1 & 1\end{array}\right)$ ;

(3)

$A = \left(\begin{array}{rrr}0 & 1 & 1 \\ -1 & 0 & 1 \\ -1 & -1 & 0\end{array}\right)$ .

Problem 1208

For any square matrix $A$ prove the identity

\det \exp A = \exp \operatorname {tr} A

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Problem 1209

Find $\cos A$ and $\sin A$ , if $A = \left(\begin{array}{rrr}1 & -4 & -3 \\ 1 & -5 & -4 \\ -1 & 5 & 4\end{array}\right)$ .

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Problem 1210

Find $\ln A$ , if $A = \left(\begin{array}{rrr}2 & -4 & -3 \\ 1 & -4 & -4 \\ -1 & 5 & 5\end{array}\right)$ .

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Problem 1211

(1)(2)

Solve the equation $X^{2} = A$ , if:

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(1)

$A = \left(\begin{array}{rrr}5 & -5 & -4 \\ 1 & 2 & -1 \\ -1 & -1 & 2\end{array}\right)$ ;

(2)

$A = \left(\begin{array}{rrr}2 & -4 & -3 \\ 1 & -4 & -4 \\ -1 & 5 & 5\end{array}\right)$ .

Problem 1212

On the space $\mathbb {C}_{n}[t]$ of polynomials of degree at most $n$ , consider the differentiation operator $\mathbf{D}$ :

(\mathbf{D} f)(x) = f^{\prime }(x)

Compute expD.

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