Math Problems Vault

Problem 28.10

Let there be, for some transformation $\varphi$ of a Euclidean space, a transformation $\varphi^{ * }$ such that $(\varphi (x), y) = \left(x, \varphi^{ * }(y)\right)$ for any vectors $x$ and $y$ . Prove that both transformations are linear.

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Problem 28.11

Find the transformation adjoint to an arbitrary transformation $\varphi$ of a one-dimensional Euclidean space.

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Problem 28.12

Find the transformation adjoint to the transformation of problem 28. 8.

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Problem 28.13

Prove that two transformations commute if and only if their adjoint transformations commute.

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Problem 28.14

Let the transformation $\varphi$ be nilpotent. Prove that its adjoint transformation is also nilpotent with the same index of nilpotency.

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Problem 28.15

Prove that for two transformations $\varphi$ and $\psi$ the product $\varphi^{ * } \psi = o$ if and only if $\operatorname {Im} \varphi$ is orthogonal to $\operatorname {Im} \psi$ .

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Problem 28.16

Let $\varphi$ be a rotation of the plane by an angle $\alpha$ . Find the adjoint transformation $\varphi^{ * }$ .

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Problem 28.17

Let $\bar{a}$ be a fixed vector of three-dimensional geometric space. The transformation $\varphi$ assigns to each vector $\bar{x}$ the cross product $\left[\bar{a}, \bar{x}\right]$ . Find $\varphi^{ * }$ .

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Problem 28.18

(1)(2)(3)

In an arithmetic space with the standard scalar product, a transformation $\varphi$ carries the vectors $a_{1}, a_{2}, a_{3}$ respectively into the vectors $b_{1}, b_{2}, b_{3}$ . Find the matrix of the adjoint transformation $\varphi^{ * }$ :

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(1)

$a_{1} = \mathbf{c}_{99}, a_{2} = \mathbf{c}_{52}, a_{3} = \mathbf{c}_{51}; b_{1} = \mathbf{c}_{110}, b_{2} = \mathbf{c}_{89}, b_{3} = \mathbf{c}_{64}$ . $\mathbf{c}_{99} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{52} = \left[\begin{smallmatrix} 1 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{51} = \left[\begin{smallmatrix} 1 \\ 1 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{110} = \left[\begin{smallmatrix} 2 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{89} = \left[\begin{smallmatrix} 1 \\ 2 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{64} = \left[\begin{smallmatrix} -1 \\ 1 \\ 2 \end{smallmatrix}\right]$ .

(2)

$a_{1} = \mathbf{c}_{142}, a_{2} = \mathbf{c}_{66}, \quad a_{3} = \mathbf{c}_{144}; \quad b_{1} = \mathbf{c}_{83}, b_{2} = \mathbf{c}_{139}, b_{3} = \mathbf{c}_{104}$ . $\mathbf{c}_{142} = \left[\begin{smallmatrix} 0 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{66} = \left[\begin{smallmatrix} 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{144} = \left[\begin{smallmatrix} 1 \\ 1 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{83} = \left[\begin{smallmatrix} 1 \\ 2 \\ 3 \end{smallmatrix}\right], \mathbf{c}_{139} = \left[\begin{smallmatrix} -2 \\ -2 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{104} = \left[\begin{smallmatrix} 1 \\ -2 \\ 3 \end{smallmatrix}\right]$ .

(3)

$a_{1} = \mathbf{c}_{57}, a_{2} = \mathbf{c}_{77}, a_{3} = \mathbf{c}_{64}; b_{1} = \mathbf{c}_{145}, b_{2} = \mathbf{c}_{101}, b_{3} = \mathbf{c}_{67}$ . $\mathbf{c}_{57} = \left[\begin{smallmatrix} 1 \\ 0 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{77} = \left[\begin{smallmatrix} 0 \\ -1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{64} = \left[\begin{smallmatrix} -1 \\ 1 \\ 2 \end{smallmatrix}\right], \mathbf{c}_{145} = \left[\begin{smallmatrix} 2 \\ 0 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{101} = \left[\begin{smallmatrix} 1 \\ -2 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{67} = \left[\begin{smallmatrix} 0 \\ 1 \\ 2 \end{smallmatrix}\right]$ .

Problem 28.19

The matrix $A$ of a transformation $\varphi$ in a basis $e$ with Gram matrix $\Gamma$ is given. Find the matrix of the adjoint transformation $\varphi^{ * }$ :

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(1)

$A = \mathbf{A}_{50}, \quad \Gamma = \mathbf{A}_{19}$ . $\mathbf{A}_{50} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & -1 \end{smallmatrix}\right], \mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right]$ .

(2)

$A = \mathbf{A}_{16}, \quad \Gamma = \mathbf{A}_{104}$ . $\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right], \mathbf{A}_{104} = \left[\begin{smallmatrix} 5 & -3 \\ -3 & 2 \end{smallmatrix}\right]$ .

(3)

$A = \mathbf{A}_{20}, \quad \Gamma = \mathbf{A}_{104}$ . $\mathbf{A}_{20} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{104} = \left[\begin{smallmatrix} 5 & -3 \\ -3 & 2 \end{smallmatrix}\right]$ .

(4)

$A = \mathbf{A}_{260}, \quad \Gamma = \mathbf{A}_{178}$ . $\mathbf{A}_{260} = \left[\begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{178} = \left[\begin{smallmatrix} 6 & 4 & 1 \\ 4 & 5 & 2 \\ 1 & 2 & 1 \end{smallmatrix}\right]$ .

(5)

$A = \mathbf{A}_{199}, \quad \Gamma = \mathbf{A}_{176}$ . $\mathbf{A}_{199} = \left[\begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{176} = \left[\begin{smallmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2 \end{smallmatrix}\right]$ .

(6)

$A = \mathbf{A}_{261}, \quad \Gamma = \mathbf{A}_{176}$ . $\mathbf{A}_{261} = \left[\begin{smallmatrix} 1 & 0 & 3 \\ 2 & 1 & 2 \\ 3 & 0 & 1 \end{smallmatrix}\right], \mathbf{A}_{176} = \left[\begin{smallmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 2 \end{smallmatrix}\right]$ .

(7)

$A = \mathbf{A}_{259}, \quad \Gamma = \mathbf{A}_{177}$ . $\mathbf{A}_{259} = \left[\begin{smallmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{smallmatrix}\right], \mathbf{A}_{177} = \left[\begin{smallmatrix} 1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 2 \end{smallmatrix}\right]$ .

(8)

$A = \mathbf{A}_{260}, \quad \Gamma = \mathbf{A}_{179}$ . $\mathbf{A}_{260} = \left[\begin{smallmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{smallmatrix}\right], \mathbf{A}_{179} = \left[\begin{smallmatrix} 3 & -2 & 1 \\ -2 & 2 & -1 \\ 1 & -1 & 1 \end{smallmatrix}\right]$ .

(9)

$A = \mathbf{A}_{269}, \quad \Gamma = \mathbf{A}_{388}$ . $\mathbf{A}_{269} = \left[\begin{smallmatrix} 0 & 2 & 3 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{smallmatrix}\right], \mathbf{A}_{388} = \left[\begin{smallmatrix} 2 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 1 & 1 \end{smallmatrix}\right]$ .

Problem 28.20

In a three-dimensional arithmetic space with the standard scalar product, find the transformation adjoint to the projection onto the line $x = 2y = 3z$ parallel to the plane $z = 0$ .

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Problem 28.21

(1)(2)

In the space $\mathcal{P}^{2}$ of polynomials of degree at most 2 with the standard scalar product, a transformation $\delta$ assigns to a polynomial its derivative. Find the adjoint transformation $\delta^{ * }$ . Write the matrix of $\delta^{ * }$ :

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(1)

in the basis $1, t, t^{2}$ ,

(2)

in the basis $1, t, \left(3t^{2} - 1\right) / 2$ .

Problem 28.22

(1)(2)

In the space $\mathcal{P}^{2}$ of polynomials of degree at most 2 with the scalar product defined in problem $25. 8, 3$ , a transformation $\delta$ assigns to a polynomial its derivative. Find the adjoint transformation $\delta^{ * }$ . Write the matrix of $\delta^{ * }$ :

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(1)

in the basis $1, t, t^{2}$ ,

(2)

in the basis $1, t, \left(3t^{2} - 1\right) / 2$ .

Problem 28.23

Let $A$ be a square matrix of order $n$ . To any square matrix $X$ of the same order the matrix $\varphi (X) = AX$ is assigned. This defines a linear transformation of the space of square matrices with the standard scalar product. Find its adjoint transformation $\varphi^{ * }$ .

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Problem 28.24

Let $A$ be a nondegenerate square matrix of order $n$ . To any square matrix $X$ of the same order the matrix $\varphi (X) = A^{ - 1} XA$ is assigned. This defines a linear transformation of the space of square matrices with the standard scalar product. Find its adjoint transformation $\varphi^{ * }$ .

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Problem 28.25

(1)(2)(3)(4)

Prove the following properties of the operation of passing to the adjoint transformation in a Euclidean space:

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(1)

$(\varphi + \psi )^{ * } = \varphi^{ * } + \psi^{ * }$

(2)

$(\varphi \psi )^{ * } = \psi^{ * } \varphi^{ * }$

(3)

$(\alpha \varphi )^{ * } = \alpha \varphi^{ * }$

(4)

If $\varphi$ has an inverse, then $\varphi^{ * }$ is also invertible, and $\left(\varphi^{ * }\right)^{ - 1} = \left(\varphi^{ - 1}\right)^{ * }$ .

Problem 28.26

(1)(2)(3)(4)

Prove that for mutually adjoint transformations of a Euclidean space the following coincide:

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(1)

the ranks,

(2)

the characteristic polynomials,

(3)

the eigenvalues,

(4)

the dimensions of the eigenspaces.

Problem 28.27

Let the transformation $\varphi$ be diagonalizable. Prove that $\varphi^{ * }$ is also diagonalizable.

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Problem 28.28

Let $e$ be a basis of eigenvectors of the transformation $\varphi$ . Prove that its biorthogonal basis $\mathbf{e}^{ * }$ consists of eigenvectors of the adjoint transformation $\varphi^{ * }$ .

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Problem 28.29

Let the transformation $\varphi$ have matrix $A$ in the basis $e$ . Prove that $\varphi^{ * }$ has matrix $A^{T}$ in the biorthogonal basis $\mathbf{e}^{ * }$ .

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Problem 28.30

(1)(2)

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(1)

Prove that the set of values of a linear transformation $\varphi$ coincides with the orthogonal complement of the kernel of the adjoint transformation $\varphi^{ * }$ .

(2)

Verify that statement 1) is equivalent to the Fredholm theorem for systems of linear equations.

Problem 28.31

Find the set of values of the transformation $\delta^{ * }$ from problem 28. 21 and directly verify that it coincides with the orthogonal complement of the kernel of $\varphi$ .

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Problem 28.32

Find the kernel of the transformation $\varphi$ from problem 28. 17 and directly verify that it coincides with the orthogonal complement of the set of values of $\varphi^{ * }$ .

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Problem 28.33

(1)(2)

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(1)

Let $\lambda_{1}$ and $\lambda_{2}$ be distinct eigenvalues of a transformation $\varphi$ of a Euclidean space. Prove that $\operatorname {Ker}\left(\varphi - \lambda_{1} \iota \right) \subseteq \operatorname {Im}\left(\varphi - \lambda_{2} \iota \right)$

(2)

Is such a statement true for a transformation of a vector space?

Problem 28.34

(1)(2)(3)

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(1)

Let the subspace $\mathcal{L} \subset \mathcal{E}$ be invariant under the transformation $\varphi$ . Prove that $\mathcal{L}^{\perp }$ is invariant under $\varphi^{ * }$ .

(2)

Let the transformation $\varphi$ have a real characteristic number. Prove that it has an $(n - 1)$ -dimensional invariant subspace. Is the converse true?

(3)

Let all roots of the characteristic polynomial of the transformation $\varphi$ be real. Prove that there exists an orthonormal basis in which the matrix of the transformation is upper triangular.

Problem 28.35

A transformation $\varphi$ is given in an orthonormal basis by the matrix $A$ . Find an orthonormal basis in which its matrix $A^{\prime }$ is upper triangular, and write the matrix $A^{\prime }$ :

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(1)

$A =\left[\begin{smallmatrix} 3 & 1 & - 1 \\ 0 & 2 & 2 \\ - 1 & - 1 & 1\end{smallmatrix}\right]$

(2)

$A = \mathbf{A}_{221}$ . $\mathbf{A}_{221} = \left[\begin{smallmatrix} 4 & 1 & -1 \\ 2 & 5 & -2 \\ 4 & 4 & -1 \end{smallmatrix}\right]$ .

(3)

$A = \mathbf{A}_{292}$ . $\mathbf{A}_{292} = \left[\begin{smallmatrix} 1 & -2 & -1 \\ -1 & 1 & 1 \\ 1 & 0 & -1 \end{smallmatrix}\right]$ .

(4)

$A = \mathbf{A}_{290}$ . $\mathbf{A}_{290} = \left[\begin{smallmatrix} 2 & 1 & -3 \\ 3 & -2 & -3 \\ 1 & 1 & -2 \end{smallmatrix}\right]$ .

(5)

$A = \mathbf{A}_{261}$ . $\mathbf{A}_{261} = \left[\begin{smallmatrix} 1 & 0 & 3 \\ 2 & 1 & 2 \\ 3 & 0 & 1 \end{smallmatrix}\right]$ .

Problem 28.36

Let $\varphi$ be a linear transformation of a Euclidean space. Prove that the root subspaces of $\varphi$ and $\varphi^{ * }$ belonging to unequal eigenvalues are orthogonal.

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Problem 28.37

Let $\varphi$ be a linear transformation of a Euclidean space. How are the Jordan forms of the transformations $\varphi$ and $\varphi^{ * }$ related?

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