Subsection 24.3

Jordan form of a matrix (24.115-24.138)

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Problem 24.115

Give an example of a matrix of order n>1n > 1 having a characteristic number λ\lambda of multiplicity k,1⩽k⩽nk, 1 \leqslant k \leqslant n, and eigenspace of dimension mm. How many Jordan blocks correspond to this λ\lambda, and what is the sum of the orders of these blocks?

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Problem 24.116

Verify by direct computation the Hamilton-Cayley theorem for the given matrix and determine its minimal polynomial:

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(1)

A37\mathbf{A}_{37}. A37=[−455−4]\mathbf{A}_{37} = \left[\begin{smallmatrix} -4 & 5 \\ 5 & -4 \end{smallmatrix}\right].

(2)

A38\mathbf{A}_{38}. A38=[53−3−1]\mathbf{A}_{38} = \left[\begin{smallmatrix} 5 & 3 \\ -3 & -1 \end{smallmatrix}\right].

(3)

A29\mathbf{A}_{29}. A29=[2363]\mathbf{A}_{29} = \left[\begin{smallmatrix} \sqrt{2} & \sqrt{3} \\ \sqrt{6} & 3 \end{smallmatrix}\right].

(4)

A82\mathbf{A}_{82}. A82=[i1−1i]\mathbf{A}_{82} = \left[\begin{smallmatrix} i & 1 \\ -1 & i \end{smallmatrix}\right].

(5)

A98\mathbf{A}_{98}. A98=[0112i]\mathbf{A}_{98} = \left[\begin{smallmatrix} 0 & 1 \\ 1 & 2 i \end{smallmatrix}\right].

(6)

A233\mathbf{A}_{233}. A233=[1−123−362−24]\mathbf{A}_{233} = \left[\begin{smallmatrix} 1 & -1 & 2 \\ 3 & -3 & 6 \\ 2 & -2 & 4 \end{smallmatrix}\right].

(7)

A222\mathbf{A}_{222}. A222=[111222−3−3−3]\mathbf{A}_{222} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ -3 & -3 & -3 \end{smallmatrix}\right].

(8)

A236\mathbf{A}_{236}. A236=[−253−2532−5−3]\mathbf{A}_{236} = \left[\begin{smallmatrix} -2 & 5 & 3 \\ -2 & 5 & 3 \\ 2 & -5 & -3 \end{smallmatrix}\right].

Problem 24.117

Can the minimal polynomial of a matrix of order nn

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(1)

be a polynomial of the first degree

(2)

have the form (t−λ)n(t - \lambda )^{n}. Give examples.

Problem 24.118
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(1)

Show that an eigenvector is a root vector. What is the height of an eigenvector?

(2)

Prove that the matrix of a linear transformation is diagonalizable if and only if the height of every root vector equals 1.

Problem 24.119

Prove that root vectors corresponding to pairwise distinct eigenvalues are linearly independent.

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Problem 24.120

Let K1\mathcal{K}_{1} and K2\mathcal{K}_{2} be root subspaces corresponding to the eigenvalues λ1≠λ2\lambda_{1} \neq \lambda_{2}. Prove that K2\mathcal{K}_{2} is invariant with respect to ψ1=φ−λ1ι\psi_{1} = \varphi - \lambda_{1} \iota, and the restriction of ψ1\psi_{1} to K2\mathcal{K}_{2} is nonsingular.

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Problem 24.121

Prove that the dimension of a root subspace equals the multiplicity of the corresponding eigenvalue as a root of the characteristic polynomial.

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Problem 24.122

Prove the linear independence of the vectors of a Jordan chain.

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Problem 24.123

Prove that the initial vectors of the Jordan chains composing a basis of a root subspace form a basis of the corresponding eigenspace.

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Problem 24.124

Let the dimension of the eigenspace corresponding to a characteristic number λ\lambda be less than the multiplicity of λ\lambda. Does every eigenvector have an associated vector?

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Problem 24.125

Find bases of the root subspaces of the linear transformation given by the matrix AA:

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(1)

A27\mathbf{A}_{27}. A27=[1−3−26]\mathbf{A}_{27} = \left[\begin{smallmatrix} 1 & -3 \\ -2 & 6 \end{smallmatrix}\right].

(2)

A11\mathbf{A}_{11}. A11=[0−213]\mathbf{A}_{11} = \left[\begin{smallmatrix} 0 & -2 \\ 1 & 3 \end{smallmatrix}\right].

(3)

A5\mathbf{A}_{5}. A5=[1−11−1]\mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right].

(4)

A51\mathbf{A}_{51}. A51=[−7−281]\mathbf{A}_{51} = \left[\begin{smallmatrix} -7 & -2 \\ 8 & 1 \end{smallmatrix}\right].

(5)

A198\mathbf{A}_{198}. A198=[−286−41064−8−4]\mathbf{A}_{198} = \left[\begin{smallmatrix} -2 & 8 & 6 \\ -4 & 10 & 6 \\ 4 & -8 & -4 \end{smallmatrix}\right].

(6)

A222\mathbf{A}_{222}. A222=[111222−3−3−3]\mathbf{A}_{222} = \left[\begin{smallmatrix} 1 & 1 & 1 \\ 2 & 2 & 2 \\ -3 & -3 & -3 \end{smallmatrix}\right].

(7)

A303\mathbf{A}_{303}. A303=[033−1862−14−10]\mathbf{A}_{303} = \left[\begin{smallmatrix} 0 & 3 & 3 \\ -1 & 8 & 6 \\ 2 & -14 & -10 \end{smallmatrix}\right].

(8)

A289\mathbf{A}_{289}. A289=[−212−102−203]\mathbf{A}_{289} = \left[\begin{smallmatrix} -2 & 1 & 2 \\ -1 & 0 & 2 \\ -2 & 0 & 3 \end{smallmatrix}\right].

Problem 24.126

Verify that the linear transformation given by the matrix AA is nilpotent and find for it a Jordan basis and the Jordan form of the matrix:

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(1)

A5\mathbf{A}_{5}. A5=[1−11−1]\mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right].

(2)

A38−2E\mathbf{A}_{38} - 2E. A38=[53−3−1]\mathbf{A}_{38} = \left[\begin{smallmatrix} 5 & 3 \\ -3 & -1 \end{smallmatrix}\right].

(3)

A52−5E\mathbf{A}_{52} - 5E. A52=[11−218−1]\mathbf{A}_{52} = \left[\begin{smallmatrix} 11 & -2 \\ 18 & -1 \end{smallmatrix}\right].

(4)

A236\mathbf{A}_{236}. A236=[−253−2532−5−3]\mathbf{A}_{236} = \left[\begin{smallmatrix} -2 & 5 & 3 \\ -2 & 5 & 3 \\ 2 & -5 & -3 \end{smallmatrix}\right].

(5)

A235\mathbf{A}_{235}. A235=[121124−1−2−3]\mathbf{A}_{235} = \left[\begin{smallmatrix} 1 & 2 & 1 \\ 1 & 2 & 4 \\ -1 & -2 & -3 \end{smallmatrix}\right].

(6)

A283+E\mathbf{A}_{283} + E. A283=[251−1−30−2−3−2]\mathbf{A}_{283} = \left[\begin{smallmatrix} 2 & 5 & 1 \\ -1 & -3 & 0 \\ -2 & -3 & -2 \end{smallmatrix}\right].

(7)

A458\mathbf{A}_{458}. A458=[100−101−1001−10100−1]\mathbf{A}_{458} = \left[\begin{smallmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{smallmatrix}\right].

(8)

A457\mathbf{A}_{457}. A457=[100−1100−101−1001−10]\mathbf{A}_{457} = \left[\begin{smallmatrix} 1 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & -1 & 0 \end{smallmatrix}\right].

(9)

A485\mathbf{A}_{485}. A485=[3−1−1−1111−311−31−13−1−1]\mathbf{A}_{485} = \left[\begin{smallmatrix} 3 & -1 & -1 & -1 \\ 1 & 1 & 1 & -3 \\ 1 & 1 & -3 & 1 \\ -1 & 3 & -1 & -1 \end{smallmatrix}\right].

Problem 24.127

Reduce the matrix to Jordan form:

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(1)

A51\mathbf{A}_{51}. A51=[−7−281]\mathbf{A}_{51} = \left[\begin{smallmatrix} -7 & -2 \\ 8 & 1 \end{smallmatrix}\right].

(2)

A30\mathbf{A}_{30}. A30=[256060144]\mathbf{A}_{30} = \left[\begin{smallmatrix} 25 & 60 \\ 60 & 144 \end{smallmatrix}\right].

(3)

A49\mathbf{A}_{49}. A49=[4−312−8]\mathbf{A}_{49} = \left[\begin{smallmatrix} 4 & -3 \\ 12 & -8 \end{smallmatrix}\right].

(4)

A52\mathbf{A}_{52}. A52=[11−218−1]\mathbf{A}_{52} = \left[\begin{smallmatrix} 11 & -2 \\ 18 & -1 \end{smallmatrix}\right].

(5)

A198\mathbf{A}_{198}. A198=[−286−41064−8−4]\mathbf{A}_{198} = \left[\begin{smallmatrix} -2 & 8 & 6 \\ -4 & 10 & 6 \\ 4 & -8 & -4 \end{smallmatrix}\right].

(6)

A299\mathbf{A}_{299}. A299=[−2−2−3236−1−2−4]\mathbf{A}_{299} = \left[\begin{smallmatrix} -2 & -2 & -3 \\ 2 & 3 & 6 \\ -1 & -2 & -4 \end{smallmatrix}\right].

(7)

A247\mathbf{A}_{247}. A247=[2−5−4−316124−20−15]\mathbf{A}_{247} = \left[\begin{smallmatrix} 2 & -5 & -4 \\ -3 & 16 & 12 \\ 4 & -20 & -15 \end{smallmatrix}\right].

(8)

A248\mathbf{A}_{248}. A248=[16032−40−8−80−16]\mathbf{A}_{248} = \left[\begin{smallmatrix} 16 & 0 & 32 \\ -4 & 0 & -8 \\ -8 & 0 & -16 \end{smallmatrix}\right].

(9)

A264\mathbf{A}_{264}. A264=[011110−101]\mathbf{A}_{264} = \left[\begin{smallmatrix} 0 & 1 & 1 \\ 1 & 1 & 0 \\ -1 & 0 & 1 \end{smallmatrix}\right].

(10)

A273\mathbf{A}_{273}. A273=[−144−10−18−2091315]\mathbf{A}_{273} = \left[\begin{smallmatrix} -1 & 4 & 4 \\ -10 & -18 & -20 \\ 9 & 13 & 15 \end{smallmatrix}\right].

(11)

A238\mathbf{A}_{238}. A238=[−310−10−74−4−2−33]\mathbf{A}_{238} = \left[\begin{smallmatrix} -3 & 10 & -10 \\ -7 & 4 & -4 \\ -2 & -3 & 3 \end{smallmatrix}\right].

(12)

(р) A283\mathbf{A}_{283}. A283=[251−1−30−2−3−2]\mathbf{A}_{283} = \left[\begin{smallmatrix} 2 & 5 & 1 \\ -1 & -3 & 0 \\ -2 & -3 & -2 \end{smallmatrix}\right].

(13)

A289\mathbf{A}_{289}. A289=[−212−102−203]\mathbf{A}_{289} = \left[\begin{smallmatrix} -2 & 1 & 2 \\ -1 & 0 & 2 \\ -2 & 0 & 3 \end{smallmatrix}\right].

(14)

A484\mathbf{A}_{484}. A484=[111111−1−11−11−11−1−11]\mathbf{A}_{484} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1 \end{smallmatrix}\right].

(15)

(р) A465\mathbf{A}_{465}. A465=[11−111−11−1−3−11−1−31−11]\mathbf{A}_{465} = \left[\begin{smallmatrix} 1 & 1 & -1 & 1 \\ 1 & -1 & 1 & -1 \\ -3 & -1 & 1 & -1 \\ -3 & 1 & -1 & 1 \end{smallmatrix}\right].

(16)

A460\mathbf{A}_{460}. A460=[0−1−2−31234000−10012]\mathbf{A}_{460} = \left[\begin{smallmatrix} 0 & -1 & -2 & -3 \\ 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 2 \end{smallmatrix}\right].

(17)

A469\mathbf{A}_{469}. A469=[−2−13−7−303−731−2731−38]\mathbf{A}_{469} = \left[\begin{smallmatrix} -2 & -1 & 3 & -7 \\ -3 & 0 & 3 & -7 \\ 3 & 1 & -2 & 7 \\ 3 & 1 & -3 & 8 \end{smallmatrix}\right].

(18)

A487\mathbf{A}_{487}. A487=[001000013400−1−100]\mathbf{A}_{487} = \left[\begin{smallmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 3 & 4 & 0 & 0 \\ -1 & -1 & 0 & 0 \end{smallmatrix}\right].

Problem 24.128

Can the following matrices of linear transformations of a real space be reduced to Jordan form? Find their Jordan form as matrices of linear transformations of a complex space:

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(1)

A45\mathbf{A}_{45}. A45=[1−13−2]\mathbf{A}_{45} = \left[\begin{smallmatrix} 1 & -1 \\ 3 & -2 \end{smallmatrix}\right].

(2)

A62\mathbf{A}_{62}. A62=[4/53/5−3/54/5]\mathbf{A}_{62} = \left[\begin{smallmatrix} 4/5 & 3/5 \\ -3/5 & 4/5 \end{smallmatrix}\right].

(3)

A44\mathbf{A}_{44}. A44=[5−141]\mathbf{A}_{44} = \left[\begin{smallmatrix} 5 & -1 \\ 4 & 1 \end{smallmatrix}\right].

(4)

A262\mathbf{A}_{262}. A262=[−1−22−2−12−3−23]\mathbf{A}_{262} = \left[\begin{smallmatrix} -1 & -2 & 2 \\ -2 & -1 & 2 \\ -3 & -2 & 3 \end{smallmatrix}\right].

(5)

A301\mathbf{A}_{301}. A301=[0−6−2−2−4−22114]\mathbf{A}_{301} = \left[\begin{smallmatrix} 0 & -6 & -2 \\ -2 & -4 & -2 \\ 2 & 11 & 4 \end{smallmatrix}\right].

(6)

A447\mathbf{A}_{447}. A447=[−12011−12001−12201−1]\mathbf{A}_{447} = \left[\begin{smallmatrix} -1 & 2 & 0 & 1 \\ 1 & -1 & 2 & 0 \\ 0 & 1 & -1 & 2 \\ 2 & 0 & 1 & -1 \end{smallmatrix}\right].

(7)

A474\mathbf{A}_{474}. A474=[−5−2−3−1422163312−340]\mathbf{A}_{474} = \left[\begin{smallmatrix} -5 & -2 & -3 & -1 \\ 4 & 2 & 2 & 1 \\ 6 & 3 & 3 & 1 \\ 2 & -3 & 4 & 0 \end{smallmatrix}\right].

Problem 24.129

Reduce to Jordan form the matrix of a linear transformation of a complex arithmetic space:

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(1)

A82\mathbf{A}_{82}. A82=[i1−1i]\mathbf{A}_{82} = \left[\begin{smallmatrix} i & 1 \\ -1 & i \end{smallmatrix}\right].

(2)

A78\mathbf{A}_{78}. A78=[ε1−1ε]\mathbf{A}_{78} = \left[\begin{smallmatrix} \varepsilon & 1 \\ -1 & \varepsilon \end{smallmatrix}\right].

(3)

A80\mathbf{A}_{80}. A80=[1−ε−ε1]\mathbf{A}_{80} = \left[\begin{smallmatrix} 1 & -\varepsilon \\ -\varepsilon & 1 \end{smallmatrix}\right].

(4)

A98\mathbf{A}_{98}. A98=[0112i]\mathbf{A}_{98} = \left[\begin{smallmatrix} 0 & 1 \\ 1 & 2 i \end{smallmatrix}\right].

Problem 24.130

Compute the value of the following polynomials of the matrix Jn(λ)J_{n}(\lambda ):

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(1)

(t−λ)m(t - \lambda )^{m}

(2)

tmt^{m}

(3)

an arbitrary polynomial f(t)f(t).

Problem 24.131

Find the Jordan form of the matrix:

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(1)

Jn2(λ)J_{n}^{2}(\lambda )

(2)

Jnm(λ)(mJ_{n}^{m}(\lambda )(m a natural number ))

(3)

Jn−1(λ)J_{n}^{ - 1}(\lambda ) (λ≠0)(\lambda \neq 0).

Problem 24.132

What can be said about a matrix AA of order nn, if its minimal polynomial satisfies the following condition:

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(1)

has all roots of multiplicity 1

(2)

coincides, up to sign, with the characteristic polynomial.

Problem 24.133

Let φm=ι\varphi^{m} = \iota for some natural number m. Prove that the Jordan form of the matrix of φ\varphi is diagonal. What numbers stand on the diagonal of this matrix?

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Problem 24.134

Find the Jordan form of the linear transformation of the complex arithmetic space with the matrix:

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(1)

A594\mathbf{A}_{594}. A594=[010…0002…0……………000…n000…0]\mathbf{A}_{594} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 2 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & n \\ 0 & 0 & 0 & \ldots & 0 \end{smallmatrix}\right].

(2)

A599\mathbf{A}_{599}. A599=[011…1002…1……………000…n000…0]\mathbf{A}_{599} = \left[\begin{smallmatrix} 0 & 1 & 1 & \ldots & 1 \\ 0 & 0 & 2 & \ldots & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & n \\ 0 & 0 & 0 & \ldots & 0 \end{smallmatrix}\right].

(3)

A609\mathbf{A}_{609}. A609=[1−10…0001−1…00………………000…1−1000…01]\mathbf{A}_{609} = \left[\begin{smallmatrix} 1 & -1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & -1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & 1 & -1 \\ 0 & 0 & 0 & \ldots & 0 & 1 \end{smallmatrix}\right].

(4)

A615\mathbf{A}_{615}. A615=[010…0001…0………….000…1123…n]\mathbf{A}_{615} = \left[\begin{smallmatrix} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & . \\ 0 & 0 & 0 & \ldots & 1 \\ 1 & 2 & 3 & \ldots & n \end{smallmatrix}\right].

(5)

A619\mathbf{A}_{619}. A619=[1−11…(−1)n−101−1…(−1)n−2………………000…1]\mathbf{A}_{619} = \left[\begin{smallmatrix} 1 & -1 & 1 & \ldots & (-1)^{n-1} \\ 0 & 1 & -1 & \ldots & (-1)^{n-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \ldots \\ 0 & 0 & 0 & \ldots & 1 \end{smallmatrix}\right].

(6)

A620\mathbf{A}_{620}. A620=[1−1…(−1)n−1−11…(−1)n−2…………………………(−1)n−1(−1)n−2…1]\mathbf{A}_{620} = \left[\begin{smallmatrix} 1 & -1 & \ldots & (-1)^{n-1} \\ -1 & 1 & \ldots & (-1)^{n-2} \\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots & \ldots & \ldots \ldots \\ (-1)^{n-1} & (-1)^{n-2} & \ldots & 1 \end{smallmatrix}\right].

(7)

A632\mathbf{A}_{632}. A632=[011…111−101…111−1−10…111…………………−1−1−1…−1−10]\mathbf{A}_{632} = \left[\begin{smallmatrix} 0 & 1 & 1 & \ldots & 1 & 1 & 1 \\ -1 & 0 & 1 & \ldots & 1 & 1 & 1 \\ -1 & -1 & 0 & \ldots & 1 & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ -1 & -1 & -1 & \ldots & -1 & -1 & 0 \end{smallmatrix}\right].

(8)

A635\mathbf{A}_{635}. A635=[0100…00−1010…000−101…00………………0000…010000…−10]\mathbf{A}_{635} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 & \ldots & 0 & 0 \\ -1 & 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & -1 & 0 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & 0 & \ldots & -1 & 0 \end{smallmatrix}\right].

(9)

A640\mathbf{A}_{640}. A640=[n−1100…000−n+1n−320…0000−n+2n−53…000…………………………………0000…−2−n+3n−10000…0−1−n+1]\mathbf{A}_{640} = \left[\begin{smallmatrix} n-1 & 1 & 0 & 0 & \ldots & 0 & 0 & 0 \\ -n+1 & n-3 & 2 & 0 & \ldots & 0 & 0 & 0 \\ 0 & -n+2 & n-5 & 3 & \ldots & 0 & 0 & 0 \\ \ldots \ldots \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \ldots \ldots \ldots \ldots \\ 0 & 0 & 0 & 0 & \ldots & -2 & -n+3 & n-1 \\ 0 & 0 & 0 & 0 & \ldots & 0 & -1 & -n+1 \end{smallmatrix}\right].

(10)

A641\mathbf{A}_{641}. A641=[123…n−1n234…n1345…12……………………n12…n−2n−1]\mathbf{A}_{641} = \left[\begin{smallmatrix} 1 & 2 & 3 & \ldots & n-1 & n \\ 2 & 3 & 4 & \ldots & n & 1 \\ 3 & 4 & 5 & \ldots & 1 & 2 \\ \ldots & \ldots & \ldots & \ldots & \ldots \ldots \ldots & \ldots \\ n & 1 & 2 & \ldots & n-2 & n-1 \end{smallmatrix}\right].

Problem 24.135

Without finding Jordan bases, determine the Jordan forms of the matrices, knowing that their characteristic polynomials equal (t−1)4(t - 1)^{4}:

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(1)

A458+E\mathbf{A}_{458} + E. A458=[100−101−1001−10100−1]\mathbf{A}_{458} = \left[\begin{smallmatrix} 1 & 0 & 0 & -1 \\ 0 & 1 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \end{smallmatrix}\right].

(2)

A460\mathbf{A}_{460}. A460=[0−1−2−31234000−10012]\mathbf{A}_{460} = \left[\begin{smallmatrix} 0 & -1 & -2 & -3 \\ 1 & 2 & 3 & 4 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 2 \end{smallmatrix}\right].

(3)

A469\mathbf{A}_{469}. A469=[−2−13−7−303−731−2731−38]\mathbf{A}_{469} = \left[\begin{smallmatrix} -2 & -1 & 3 & -7 \\ -3 & 0 & 3 & -7 \\ 3 & 1 & -2 & 7 \\ 3 & 1 & -3 & 8 \end{smallmatrix}\right].

Problem 24.136

Verify that the matrices A221\mathbf{A}_{221} and −A273- \mathbf{A}_{273} have the same characteristic polynomials. Find their minimal polynomials and Jordan forms. A221=[41−125−244−1],A273=[−144−10−18−2091315]\mathbf{A}_{221} = \left[\begin{smallmatrix} 4 & 1 & -1 \\ 2 & 5 & -2 \\ 4 & 4 & -1 \end{smallmatrix}\right], \mathbf{A}_{273} = \left[\begin{smallmatrix} -1 & 4 & 4 \\ -10 & -18 & -20 \\ 9 & 13 & 15 \end{smallmatrix}\right].

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Problem 24.137

Determine the Jordan form of the matrix AA from the given characteristic polynomial p(t)=−(t+1)3(t−−2)2p(t) = - (t + 1)^{3}(t - - 2)^{2}, knowing that Rg⁡(A−2E)=3,arg⁡(A+E)=4\operatorname {Rg}(A - 2E) = 3, \arg (A + E) = 4.

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Problem 24.138

Find the exponential of the matrix:

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(1)

Jn(0)J_{n}(0)

(2)

A5\mathbf{A}_{5}. A5=[1−11−1]\mathbf{A}_{5} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & -1 \end{smallmatrix}\right].

(3)

[0−αα0],α∈R\left[\begin{smallmatrix} 0 & - \alpha \\ \alpha & 0\end{smallmatrix}\right], \quad \alpha \in \mathbb {R}.