All textbooks/Collection of problems in analytical geometry and linear algebra/12 · Functions on a linear space

Section 32

Bilinear and quadratic functions

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32.1 · Bilinear and quadratic functions in a real linear space (32.1-32.26)

32.2 · Quadratic functions in Euclidean space. Pairs of forms (32.27-32.39)

32.3 · Bilinear and quadratic functions in complex space (32.40-32.47)

Subsection 32.1Bilinear and quadratic functions in a real linear space (32.1-32.26)[26/65%]

Problem 32.1

(1)(2)(3)(4)(5)(6)(7)

Form the matrix of the given bilinear form and write the corresponding quadratic form in an $n$ -dimensional vector space:

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(1)

$x_{1} y_{1} \quad (n = 1)$ ,

(2)

$x_{1} y_{1} \quad (n = 2)$

(3)

$2x_{1} y_{1} - x_{1} y_{2} - x_{2} y_{1} - 5x_{2} y_{2} \quad (n = 2)$

(4)

$x_{1} y_{2} - 3x_{1} y_{3} + 7x_{2} y_{3} + x_{2} y_{1} - 3x_{3} y_{1} + 7x_{3} y_{2} + x_{3} y_{3} \quad (n = 3)$

(5)

$\sum_{i = 1}^{n} x_{i} y_{i}$

(6)

$\sum_{i = 1}^{n} x_{i} y_{n - i + 1}$

(7)

$\sum_{\left|i - j\right| \leqslant 1} x_{i} y_{j}$

Problem 32.2

(1)(2)(3)(4)(5)

Recover the symmetric bilinear form in an $n$ -dimensional vector space from the given quadratic form and form its matrix:

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(1)

$- 3x_{1}^{2} \quad (n = 1)$

(2)

$- 18x_{1} x_{2} + 9x_{2}^{2} \quad (n = 2)$

(3)

$x_{1}^{2} + 4x_{1} x_{2} + 4x_{1} x_{3} + 5x_{2}^{2} + 12x_{2} x_{3} + 7x_{3}^{2} \quad (n = 3)$

(4)

$2x_{1}^{2} - 6x_{1} x_{2} - 3x_{2}^{2} \quad (n = 3)$

(5)

$\sum_{i = 1}^{n - 1} x_{i} x_{i + 1}$

Problem 32.3

(1)(2)(3)(4)(5)(6)(7)(8)

Write the quadratic form having the given matrix:

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(1)

$\mathbf{A}_{47}$ . $\mathbf{A}_{47} = \left[\begin{smallmatrix} 5 & -2 \\ -2 & 8 \end{smallmatrix}\right]$ .

(2)

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

(3)

$\mathbf{A}_{307}$ . $\mathbf{A}_{307} = \left[\begin{smallmatrix} 4 & 2 & 4 \\ 2 & -3 & 0 \\ 4 & 0 & 4 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{280}$ . $\mathbf{A}_{280} = \left[\begin{smallmatrix} 0 & 2 & 1 \\ 2 & 8 & 2 \\ 1 & 2 & 0 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{484}$ . $\mathbf{A}_{484} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 1 & -1 \\ 1 & -1 & -1 & 1 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{471}$ . $\mathbf{A}_{471} = \left[\begin{smallmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & -1 \\ 0 & -1 & 2 & 0 \\ 0 & -1 & 0 & 2 \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{593}$ . $\mathbf{A}_{593} = \left[\begin{smallmatrix} 2 & 0 & -1 & 0 & 0 & 0 \\ 0 & 2 & 0 & -1 & 0 & 0 \\ -1 & 0 & 2 & -1 & 0 & 0 \\ 0 & -1 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 \\ 0 & 0 & 0 & 0 & -1 & 2 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{634}$ . $\mathbf{A}_{634} = \left[\begin{smallmatrix} 0 & 1 & 0 & 0 & \ldots & 0 & 0 \\ 1 & 0 & 1 & 0 & \ldots & 0 & 0 \\ 0 & 1 & 0 & 1 & \ldots & 0 & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & 0 & \ldots & 0 & 1 \\ 0 & 0 & 0 & 0 & \ldots & 1 & 0 \end{smallmatrix}\right]$ .

Problem 32.4

(1)(2)

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(1)

Recover the symmetric bilinear function from the quadratic function it generates.

(2)

Prove that any bilinear function $\mathrm{b}(x, y)$ can be uniquely represented as a sum $\mathrm{b}(x, y) = \mathrm{b}_{ + }(x, y) + \mathrm{b}_{ - }(x, y)$ , where $\mathrm{b}_{ + }(x, y) = \mathrm{b}_{ + }(y, x)$ , and $\mathrm{b}_{ - }(x, y) = - \mathrm{b}_{ - }(y, x)$ . Prove that then $\mathrm{b}(x, x) = \mathrm{b}_{ + }(x, x)$ .

Problem 32.5

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

How will the matrix of a bilinear (quadratic) function change if the basis $e_{1}, \ldots , e_{n}$ is changed as follows:

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(1)

swap the $i$ -th and $j$ -th basis vectors

(2)

multiply the $i$ -th basis vector by a number $\lambda \neq 0$

(3)

replace the vector $e_{i}$ by $e_{i} + \lambda e_{j}(i \neq j)$

(4)

arrange the basis vectors in reverse order?

Problem 32.6

A quadratic function and a linear transformation have the same matrices in some basis. What must the transition matrix from this basis to another basis be, in order that the matrices of the quadratic function and the linear transformation coincide in the other basis as well?

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Problem 32.7

(1)(2)(3)(4)(5)(6)(7)

A quadratic function is given in the basis $e_{1}, \ldots , e_{n}$ . Write this quadratic function in the basis $e_{1}^{\prime }, \ldots , e_{n}^{\prime }$ :

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(1)

$25x_{1}^{2} - 14x_{1} x_{2} + 2x_{2}^{2}, \quad e_{1}^{\prime } = e_{1} + e_{2}, \quad e_{2}^{\prime } = - e_{1} + e_{2}$

(2)

$3x_{1}^{2} + 10x_{1} x_{2} + 9x_{2}^{2}, \quad e_{1}^{\prime } = 2e_{1} - e_{2}, \quad e_{2}^{\prime } = e_{1} - e_{2}$

(3)

$4x_{1}^{2} - 12x_{1} x_{2} + 9x_{2}^{2}, \quad e_{1}^{\prime } = \frac{1}{4} e_{1} - \frac{1}{6} e_{2}, \quad e_{2}^{\prime } = \frac{1}{4} e_{1} + \frac{1}{6} e_{2}$

(4)

$x_{1}^{2} + 4x_{1} x_{2} + 4x_{1} x_{3} - x_{3}^{2}, \quad e_{1}^{\prime } = e_{1} + e_{2} + e_{3}, \quad e_{2}^{\prime } = 2e_{1} - - e_{2} + e_{3}, \quad e_{3}^{\prime } = - e_{1} + 2e_{2} - 3e_{3}$

(5)

$x_{1}^{2} + 2x_{1} x_{2} - x_{1} x_{3} - x_{2}^{2} + 2x_{2} x_{3} + x_{3}^{2}, \quad e_{1}^{\prime } = 2e_{1} - e_{3}, \quad e_{2}^{\prime } = - e_{1} + 2e_{2} - e_{3}, \quad e_{3}^{\prime } = - e_{2} + e_{3}$

(6)

$5x_{1}^{2} + 5x_{2}^{2} + 3x_{3}^{2} + 2x_{1} x_{2} + 2\sqrt{2} x_{1} x_{3} + 2\sqrt{2} x_{2} x_{3}, e_{1}^{\prime } = e_{1} + e_{2} - 2\sqrt{2} e_{3}, \quad e_{2}^{\prime } = e_{1} - e_{2}, \quad e_{3}^{\prime } = \sqrt{2} e_{1} + \sqrt{2} e_{2} + e_{3}$

(7)

$\sum_{i = 1}^{n - 1} x_{i} x_{i + 1}, \quad e_{i}^{\prime } = e_{i} + e_{i + 1} + \ldots + e_{n}, \quad i = 1, 2, \ldots , n$ .

Problem 32.8

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)

Reduce the given quadratic form to canonical form using Lagrange's method or elementary transformations of its matrix. Find the rank, the positive and negative indices of inertia, and the signature of this form:

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(1)

$4x_{1}^{2} + 4x_{1} x_{2} + 5x_{2}^{2}$

(2)

$x_{1}^{2} - x_{1} x_{2} - x_{2}^{2}$

(3)

$- x_{1} x_{2}$

(4)

$25x_{1}^{2} + 30x_{1} x_{2} + 9x_{2}^{2}$

(5)

$2x_{1} x_{2} - x_{1}^{2} - 2x_{2}^{2}$

(6)

$24x_{1} x_{2} - 16x_{1}^{2} - 9x_{2}^{2}$

(7)

$x_{1}^{2} + 4x_{1} x_{3} + x_{2}^{2} + 2x_{2} x_{3} + 4x_{3}^{2}$

(8)

$x_{1}^{2} + 2x_{1} x_{2} + 2x_{1} x_{3} - 3x_{2}^{2} - 6x_{2} x_{3} - 4x_{3}^{2}$

(9)

$2x_{1}^{2} + 8x_{1} x_{2} + 4x_{1} x_{3} + 9x_{2}^{2} + 19x_{3}^{2}$

(10)

$9x_{1}^{2} - 12x_{1} x_{2} - 6x_{1} x_{3} + 4x_{2}^{2} + 4x_{2} x_{3} + x_{3}^{2}$

(11)

$8x_{1}^{2} + 8x_{2}^{2} + x_{3}^{2} + 16x_{1} x_{2} + 4x_{1} x_{3} + 4x_{2} x_{3}$

(12)

(р) $x_{1} x_{2} + x_{2} x_{3} + x_{1} x_{3}$

(13)

$x_{1}^{2} + 2x_{2}^{2} + 2x_{3}^{2} + 3x_{4}^{2} + 2x_{1} x_{2} + 2x_{2} x_{3} + 2x_{3} x_{4}$

(14)

$x_{1}^{2} - 2x_{1} x_{3} + x_{2}^{2} - 2x_{2} x_{4} + x_{3}^{2} - 2x_{3} x_{5} + x_{4}^{2} - 2x_{4} x_{6} + x_{5}^{2} + x_{6}^{2}$

(15)

$x_{1} x_{2} + 2x_{2} x_{3} - 3x_{3} x_{4}$

(16)

$x_{1} x_{2} + x_{2} x_{3} - x_{1}^{2} - x_{2}^{2} - x_{3}^{2}$ .

Problem 32.9

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)

Determine which quadratic forms from problem 32.8 are positive definite, negative definite, semidefinite:

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(1)

$4x_{1}^{2} + 4x_{1} x_{2} + 5x_{2}^{2}$

(2)

$x_{1}^{2} - x_{1} x_{2} - x_{2}^{2}$

(3)

$- x_{1} x_{2}$

(4)

$25x_{1}^{2} + 30x_{1} x_{2} + 9x_{2}^{2}$

(5)

$2x_{1} x_{2} - x_{1}^{2} - 2x_{2}^{2}$

(6)

$24x_{1} x_{2} - 16x_{1}^{2} - 9x_{2}^{2}$

(7)

$x_{1}^{2} + 4x_{1} x_{3} + x_{2}^{2} + 2x_{2} x_{3} + 4x_{3}^{2}$

(8)

$x_{1}^{2} + 2x_{1} x_{2} + 2x_{1} x_{3} - 3x_{2}^{2} - 6x_{2} x_{3} - 4x_{3}^{2}$

(9)

$2x_{1}^{2} + 8x_{1} x_{2} + 4x_{1} x_{3} + 9x_{2}^{2} + 19x_{3}^{2}$

(10)

$9x_{1}^{2} - 12x_{1} x_{2} - 6x_{1} x_{3} + 4x_{2}^{2} + 4x_{2} x_{3} + x_{3}^{2}$

(11)

$8x_{1}^{2} + 8x_{2}^{2} + x_{3}^{2} + 16x_{1} x_{2} + 4x_{1} x_{3} + 4x_{2} x_{3}$

(12)

(р) $x_{1} x_{2} + x_{2} x_{3} + x_{1} x_{3}$

(13)

$x_{1}^{2} + 2x_{2}^{2} + 2x_{3}^{2} + 3x_{4}^{2} + 2x_{1} x_{2} + 2x_{2} x_{3} + 2x_{3} x_{4}$

(14)

$x_{1}^{2} - 2x_{1} x_{3} + x_{2}^{2} - 2x_{2} x_{4} + x_{3}^{2} - 2x_{3} x_{5} + x_{4}^{2} - 2x_{4} x_{6} + x_{5}^{2} + x_{6}^{2}$

(15)

$x_{1} x_{2} + 2x_{2} x_{3} - 3x_{3} x_{4}$

(16)

$x_{1} x_{2} + x_{2} x_{3} - x_{1}^{2} - x_{2}^{2} - x_{3}^{2}$ .

Problem 32.10

(1)(2)(3)(4)(5)(6)(7)

Reduce the given bilinear form to canonical form:

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(1)

$x_{1} y_{1} + x_{1} y_{2} + x_{2} y_{1} + 3x_{2} y_{2}$

(2)

$x_{1} y_{1} - x_{1} y_{2} - x_{2} y_{1} + x_{2} y_{2}$

(3)

$13x_{1} y_{1} - 5x_{1} y_{2} - 5x_{2} y_{1} + 2x_{2} y_{2}$

(4)

$- x_{1} y_{2} - x_{2} y_{1} + x_{2} y_{2}$

(5)

$x_{1} y_{2} + x_{2} y_{1} + x_{1} y_{3} + x_{3} y_{1} + x_{2} y_{3} + x_{3} y_{2}$

(6)

$x_{1} y_{1} + 2x_{2} y_{2} + 3x_{3} y_{3} + x_{1} y_{2} + x_{2} y_{1} + x_{1} y_{3} + x_{3} y_{1} + 2x_{2} y_{3} + 2x_{3} y_{2}$

(7)

$x_{1} y_{1} + x_{1} y_{2} + x_{2} y_{1} + x_{2} y_{3}$ .

Problem 32.11

Prove that a nonsymmetric bilinear function cannot be reduced to diagonal form.

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Problem 32.12

(1)(2)(3)(4)(5)

Reduce a quadratic form depending on a real parameter $\lambda$ to canonical form for all possible values of $\lambda$ :

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(1)

$3x_{1}^{2} - 2x_{1} x_{2} + \lambda x_{2}^{2}$

(2)

$8x_{1}^{2} + \lambda x_{1} x_{2} + 2x_{2}^{2}$

(3)

$2x_{1}^{2} + 8x_{1} x_{2} + 4x_{1} x_{3} + 6x_{2}^{2} + \lambda x_{3}^{2}$

(4)

$x_{1}^{2} + x_{2}^{2} + 4x_{3}^{2} + \lambda x_{4}^{2} + 4x_{1} x_{3} + 2x_{1} x_{4} + 2x_{2} x_{3} + 2x_{2} x_{4} + 5x_{3} x_{4}$

(5)

$3x_{2}^{2} + 6x_{1} x_{2} + 2x_{1} x_{3} + 4x_{2} x_{3} + \lambda x_{2} x_{4} + x_{3}^{2} + x_{3} x_{4} + x_{4}^{2}$ .

Problem 32.13

(1)(2)(3)(4)(5)(6)

Reduce the given quadratic form to canonical form in an $n$ -dimensional space:

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(1)

$x_{1}^{2} + 2\sum_{i = 2}^{n} x_{i}^{2} - 2\sum_{i = 1}^{n - 1} x_{i} x_{i + 1}$

(2)

$x_{1}^{2} + 2\sum_{i = 1}^{n - 1}(- 1)^{i} x_{i} x_{i + 1}$

(3)

$\sum_{i = 1}^{n} x_{i}^{2} + \sum_{1 \leqslant i < j \leqslant n} x_{i} x_{j}$

(4)

$\sum_{1 \leqslant i < j \leqslant n} x_{i} x_{j}$

(5)

$- \sum_{i = 1}^{n} ix_{i}^{2} + 2\sum_{1 \leqslant i < j \leqslant n} ix_{i} x_{j}$

(6)

$\sum_{i = 1}^{n}\left((i - 1)^{2} + 1\right) x_{i}^{2} + 2\sum_{1 \leqslant i < j \leqslant n} ix_{i} x_{j}$ .

Problem 32.14

(1)(2)(3)

Prove that for the positive definiteness of a quadratic function $\mathrm{k}(x)$ any one of the following conditions is necessary and sufficient:

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(1)

$\mathrm{k}\left(e_{i}\right) > 0(i = 1, \ldots , n)$ for any basis $e_{1}, \ldots , e_{n}$

(2)

$\mathrm{k}(x)$ reduces to diagonal form with positive coefficients

(3)

$\mathrm{k}(x)$ reduces to the canonical form $\xi_{1}^{2} + \ldots + \xi_{n}^{2}$ .

Problem 32.15

Show that for the positive definiteness of a quadratic function, the positivity of all diagonal elements of its matrix in some basis is necessary but not sufficient.

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Problem 32.16

Prove that a quadratic form is negative definite if and only if the signs of the principal minors of its matrix alternate as follows:

\Delta _{1} < 0, \Delta _{2} > 0, \Delta _{3} < 0, \ldots , \operatorname {sign} \Delta _{n} = (- 1)^{n}.

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Problem 32.17

(1)(2)

Let the rank of a quadratic function $\mathrm{k}(x)$ in an $n$ -dimensional vector space $\mathcal{L}$ equal $r$ . Prove the statements:

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(1)

in $\mathcal{L}$ there exists a basis in which the principal minors $\Delta_{k}$ of the matrix of the function $\mathrm{k}(x)$ are nonzero for $k = 1, \ldots , r$ and equal zero for $k = r + 1, \ldots , n$ .

(2)

let in some basis the principal minors $\Delta_{k}(k = 1, \ldots , r)$ of the matrix of the function $\mathrm{k}(x)$ be nonzero. Then $\mathrm{k}(x)$ reduces to the diagonal form $\sum_{k = 1}^{r} \frac{\Delta_{k}}{\Delta_{k - 1}} \xi_{k}^{2}\left(\Delta_{0} = 1\right)$ and to the canonical form $\sum_{k = 1}^{r} \varepsilon_{k} \xi_{k}^{2}$ , where $\varepsilon_{k} = \operatorname {sign} \frac{\Delta_{k}}{\Delta_{k - 1}}(k = 1, \ldots , r)$ .

Problem 32.18

(1)(2)(3)(4)(5)

For which values of the parameter $\lambda$ is the given quadratic form positive definite, negative definite, or semidefinite:

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(1)

$\lambda x_{1}^{2} - 4x_{1} x_{2} + (\lambda + 3) x_{2}^{2}$

(2)

$- 9x_{1}^{2} + 6\lambda x_{1} x_{2} - x_{2}^{2}$

(3)

$\lambda x_{1}^{2} + 8x_{2}^{2} + x_{3}^{2} + 16x_{1} x_{2} + 4x_{1} x_{3} + 4x_{2} x_{3}$

(4)

$x_{1}^{2} + 2\lambda x_{1} x_{2} + 2x_{1} x_{3} + 4x_{2}^{2} - \lambda x_{3}^{2} + 2x_{2} x_{3}$

(5)

$(4 - \lambda ) x_{1}^{2} + (4 - \lambda ) x_{2}^{2} - (2 + \lambda ) x_{3}^{2} + 4x_{1} x_{2} - 8x_{1} x_{3} + 8x_{2} x_{3}$ ?

Problem 32.19

Let $\mathrm{k}(x)$ be a quadratic function on a vector space $\mathcal{L}$ . Is the set of all vectors $x$ from $\mathcal{L}$ for which $\mathrm{k}(x) \geqslant 0(\mathrm{k}(x) \leqslant 0)$ a linear subspace of $\mathcal{L}$ ? Consider the examples $\mathrm{k}(x) = x_{1}^{2} + x_{2}^{2} - x_{3}^{2}(n = 3)$ and $\mathrm{h}(x) = x_{1}^{2} + x_{2}^{2}(n = 3)$ .

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Problem 32.20

(1)(2)

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(1)

In the matrix of a positive definite quadratic form, one diagonal element was increased. Prove that the determinant of the matrix increased.

(2)

Prove that in the matrix of a positive definite quadratic form the largest element in absolute value is positive.

Problem 32.21

(1)(2)

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(1)

Prove that in the vector space of real square matrices of order $n$ the function $\mathrm{k}(X) = \operatorname {tr}\left(X^{T} X\right)$ is a positive definite quadratic function.

(2)

Prove that in the vector space of real square matrices of order $n$ the function $\mathrm{k}(X) = \operatorname {tr}\left(X^{2}\right)$ is a quadratic function. Find its rank and signature.

Problem 32.22

Prove that the function

I(f, g) = \int _{ - 1}^{1} f(t) g(t) dt

is a symmetric bilinear function in the space of polynomials of degree at most $n$ . Reduce it to canonical form for $n = 3$ .

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Problem 32.23

Prove that the rank of a bilinear function equals 1 if and only if this function is the product of two nonzero linear functions.

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Problem 32.24

Prove that for a quadratic function to be representable as the product of two real linear functions, it is necessary and sufficient that either the rank of this quadratic function not exceed 1, or the rank equal 2 with signature equal to zero.

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Problem 32.25

Under what necessary and sufficient condition can the quadratic functions $\mathrm{k}(x)$ and $- \mathrm{k}(x)$ be reduced to one and the same canonical form?

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Problem 32.26

(1)(2)

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(1)

Let $\mathrm{b}(x, y)$ be a bilinear function on a vector space $\mathcal{L}$ . Call the function $\mathrm{b}(x, y)$ invariant with respect to a linear transformation $\varphi$ of the space $\mathcal{L}$ if $\mathrm{b}(\varphi (x), \varphi (y)) = \mathrm{b}(x, y)$ for all $x, y\in \mathcal{L}$ . Prove that $\mathrm{b}$ is invariant with respect to $\varphi$ if and only if their matrices ( $B$ and $A$ , respectively) in some basis satisfy the equality $A^{T} BA = B$ .

(2)

Find all linear transformations of a two-dimensional space with respect to which the bilinear form is invariant:

(1) $x_{1} y_{1} + x_{2} y_{2}$

(2) $x_{1} y_{1} - x_{2} y_{2}$

Subsection 32.2Quadratic functions in Euclidean space. Pairs of forms (32.27-32.39)[13/62%]

Problem 32.27

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

A quadratic (bilinear) function is written in an orthonormal basis of an $n$ -dimensional Euclidean space. Find an orthonormal basis in which the given function has diagonal form, and write this diagonal form.

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(1)

$n = 2$ , $- 4x_{1}^{2} + 10x_{1} x_{2} - 4x_{2}^{2}$

(2)

$n = 2$ , $\frac{4}{3} x_{1}^{2} - 2x_{1} x_{2} + \frac{3}{4} x_{2}^{2}$

(3)

$n = 2$ , $7x_{1}^{2} + 4\sqrt{3} x_{1} x_{2} + 3x_{2}^{2}$

(4)

$n = 2$ , $- x_{1} y_{1} + 3x_{1} y_{2} + 3x_{2} y_{1} - 9x_{2} y_{2}$

(5)

$n = 2$ , $- x_{1} y_{1} + \frac{1}{2} x_{1} y_{2} + \frac{1}{2} x_{2} y_{1} - x_{2} y_{2} n = 3:$

(6)

$n = 3$ , $- x_{1}^{2} + x_{1} x_{2} - x_{2}^{2}$

(7)

$n = 3$ , $2x_{1}^{2} - 4x_{1} x_{2} + 9x_{2}^{2} + 4x_{2} x_{3} + 2x_{3}^{2}$

(8)

$n = 3$ , $x_{1} y_{1} - x_{1} y_{2} - x_{2} y_{1} + 2x_{2} y_{2} - x_{2} y_{3} - x_{3} y_{2} + x_{3} y_{3}$

(9)

$n = 3$ , $2x_{1} y_{2} + 2x_{2} y_{1} - 2x_{1} y_{3} - 2x_{3} y_{1} + 4x_{2} y_{2} + 4x_{2} y_{3} + 4x_{3} y_{2} - 3x_{3} y_{3}$

(10)

$n = 3$ , (р) $2x_{1}^{2} + 4x_{1} x_{2} - 2x_{1} x_{3} - x_{2}^{2} + 4x_{2} x_{3} + 2x_{3}^{2}$

(11)

$n = 3$ , $3x_{1}^{2} - 2x_{1} x_{2} - 2x_{1} x_{3} + 3x_{2}^{2} - 2x_{2} x_{3} + 3x_{3}^{2}$

(12)

$n = 3$ , $3x_{1}^{2} + 8x_{1} x_{2} - 8x_{1} x_{3} - 7x_{2}^{2} - 8x_{2} x_{3} + 3x_{3}^{2}$

(13)

$n = 3$ , $x_{1}^{2} - x_{1} x_{2} + x_{1} x_{3} + x_{2}^{2} + x_{2} x_{3} + x_{3}^{2}$

(14)

$n = 3$ , $4x_{1}^{2} + 4x_{1} x_{2} - 12x_{1} x_{3} + x_{2}^{2} - 6x_{2} x_{3} + 9x_{3}^{2}$

(15)

$n = 3$ , $x_{1} y_{2} + x_{2} y_{1} - 2x_{1} y_{3} - 2x_{3} y_{1} - x_{1} y_{1} - x_{2} y_{2} + 2x_{2} y_{3} + 2x_{3} y_{2} - 4x_{3} y_{3}$

(16)

$n = 3$ , $x_{1} y_{2} + x_{2} y_{1} + x_{1} y_{3} + x_{3} y_{1} + x_{2} y_{2} - x_{3} y_{3} n = 4$ :

(17)

$n = 4$ , $x_{1}^{2} + 2x_{1} x_{2} + 2x_{1} x_{3} + 2x_{1} x_{4} + x_{2}^{2} - 2x_{2} x_{3} - 2x_{2} x_{4} + x_{3}^{2} - - 2x_{3} x_{4} + x_{4}^{2}$

(18)

$n = 4$ , $x_{1}^{2} - 2x_{1} x_{2} + 6x_{1} x_{3} + 8x_{1} x_{4} + 4x_{2}^{2} - 2x_{2} x_{4} - 6x_{3} x_{4} + x_{4}^{2}$

(19)

$n = 4$ , $2x_{1}^{2} - 4x_{1} x_{2} + 4x_{1} x_{3} + 6x_{2}^{2} - 6x_{2} x_{3} + 2x_{2} x_{4} + 6x_{3}^{2} + 2x_{3} x_{4} + 4x_{4}^{2}$

(20)

$n = 4$ , $x_{1}^{2} + 2x_{1} x_{2} + 2x_{1} x_{3} + 2x_{1} x_{4} - x_{2}^{2} - x_{3}^{2} - x_{4}^{2}$

(21)

$n = 4$ , $\frac{1}{2} x_{1} y_{2} + \frac{1}{2} x_{2} y_{1} + \frac{1}{2} x_{3} y_{4} + \frac{1}{2} x_{4} y_{3}$

(22)

$n = 4$ , $3x_{1}^{2} - 8x_{1} x_{2} - 3x_{2}^{2} - x_{3}^{2} + 4x_{3} x_{4} - 4x_{4}^{2}$

(23)

$n = 4$ , $\sum_{i = 1}^{n} x_{i}^{2} + \sum_{1 \leqslant i < j \leqslant n} x_{i} x_{j}$

(24)

$n = 4$ , $\sum_{i, j = 1}^{n}(- 1)^{i + j} x_{i} y_{j}$

(25)

$n = 4$ , $\sum_{i = 1}^{2n - 1} x_{i} x_{2n - i}$

(26)

$n = 4$ , $\sum_{i = 1}^{2n} x_{i}^{2} + \sum_{i = 1}^{2n} x_{i} x_{2n - i + 1}$

(27)

$n = 4$ , $\sum_{i = 1}^{n - 1} x_{i} x_{i + 1}$ .

Problem 32.28

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)

Find the canonical form, rank, and signature of each of the quadratic and bilinear forms of problem 32. 27:

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(1)

$n = 2$ , $- 4x_{1}^{2} + 10x_{1} x_{2} - 4x_{2}^{2}$

(2)

$n = 2$ , $\frac{4}{3} x_{1}^{2} - 2x_{1} x_{2} + \frac{3}{4} x_{2}^{2}$

(3)

$n = 2$ , $7x_{1}^{2} + 4\sqrt{3} x_{1} x_{2} + 3x_{2}^{2}$

(4)

$n = 2$ , $- x_{1} y_{1} + 3x_{1} y_{2} + 3x_{2} y_{1} - 9x_{2} y_{2}$

(5)

$n = 2$ , $- x_{1} y_{1} + \frac{1}{2} x_{1} y_{2} + \frac{1}{2} x_{2} y_{1} - x_{2} y_{2} n = 3:$

(6)

$n = 3$ , $- x_{1}^{2} + x_{1} x_{2} - x_{2}^{2}$

(7)

$n = 3$ , $2x_{1}^{2} - 4x_{1} x_{2} + 9x_{2}^{2} + 4x_{2} x_{3} + 2x_{3}^{2}$

(8)

$n = 3$ , $x_{1} y_{1} - x_{1} y_{2} - x_{2} y_{1} + 2x_{2} y_{2} - x_{2} y_{3} - x_{3} y_{2} + x_{3} y_{3}$

(9)

$n = 3$ , $2x_{1} y_{2} + 2x_{2} y_{1} - 2x_{1} y_{3} - 2x_{3} y_{1} + 4x_{2} y_{2} + 4x_{2} y_{3} + 4x_{3} y_{2} - 3x_{3} y_{3}$

(10)

$n = 3$ , (р) $2x_{1}^{2} + 4x_{1} x_{2} - 2x_{1} x_{3} - x_{2}^{2} + 4x_{2} x_{3} + 2x_{3}^{2}$

(11)

$n = 3$ , $3x_{1}^{2} - 2x_{1} x_{2} - 2x_{1} x_{3} + 3x_{2}^{2} - 2x_{2} x_{3} + 3x_{3}^{2}$

(12)

$n = 3$ , $3x_{1}^{2} + 8x_{1} x_{2} - 8x_{1} x_{3} - 7x_{2}^{2} - 8x_{2} x_{3} + 3x_{3}^{2}$

(13)

$n = 3$ , $x_{1}^{2} - x_{1} x_{2} + x_{1} x_{3} + x_{2}^{2} + x_{2} x_{3} + x_{3}^{2}$

(14)

$n = 3$ , $4x_{1}^{2} + 4x_{1} x_{2} - 12x_{1} x_{3} + x_{2}^{2} - 6x_{2} x_{3} + 9x_{3}^{2}$

(15)

$n = 3$ , $x_{1} y_{2} + x_{2} y_{1} - 2x_{1} y_{3} - 2x_{3} y_{1} - x_{1} y_{1} - x_{2} y_{2} + 2x_{2} y_{3} + 2x_{3} y_{2} - 4x_{3} y_{3}$

(16)

$n = 3$ , $x_{1} y_{2} + x_{2} y_{1} + x_{1} y_{3} + x_{3} y_{1} + x_{2} y_{2} - x_{3} y_{3} n = 4$ :

(17)

$n = 4$ , $x_{1}^{2} + 2x_{1} x_{2} + 2x_{1} x_{3} + 2x_{1} x_{4} + x_{2}^{2} - 2x_{2} x_{3} - 2x_{2} x_{4} + x_{3}^{2} - - 2x_{3} x_{4} + x_{4}^{2}$

(18)

$n = 4$ , $x_{1}^{2} - 2x_{1} x_{2} + 6x_{1} x_{3} + 8x_{1} x_{4} + 4x_{2}^{2} - 2x_{2} x_{4} - 6x_{3} x_{4} + x_{4}^{2}$

(19)

$n = 4$ , $2x_{1}^{2} - 4x_{1} x_{2} + 4x_{1} x_{3} + 6x_{2}^{2} - 6x_{2} x_{3} + 2x_{2} x_{4} + 6x_{3}^{2} + 2x_{3} x_{4} + 4x_{4}^{2}$

(20)

$n = 4$ , $x_{1}^{2} + 2x_{1} x_{2} + 2x_{1} x_{3} + 2x_{1} x_{4} - x_{2}^{2} - x_{3}^{2} - x_{4}^{2}$

(21)

$n = 4$ , $\frac{1}{2} x_{1} y_{2} + \frac{1}{2} x_{2} y_{1} + \frac{1}{2} x_{3} y_{4} + \frac{1}{2} x_{4} y_{3}$

(22)

$n = 4$ , $3x_{1}^{2} - 8x_{1} x_{2} - 3x_{2}^{2} - x_{3}^{2} + 4x_{3} x_{4} - 4x_{4}^{2}$

(23)

$n = 4$ , $\sum_{i = 1}^{n} x_{i}^{2} + \sum_{1 \leqslant i < j \leqslant n} x_{i} x_{j}$

(24)

$n = 4$ , $\sum_{i, j = 1}^{n}(- 1)^{i + j} x_{i} y_{j}$

(25)

$n = 4$ , $\sum_{i = 1}^{2n - 1} x_{i} x_{2n - i}$

(26)

$n = 4$ , $\sum_{i = 1}^{2n} x_{i}^{2} + \sum_{i = 1}^{2n} x_{i} x_{2n - i + 1}$

(27)

$n = 4$ , $\sum_{i = 1}^{n - 1} x_{i} x_{i + 1}$ .

Problem 32.29

Prove that a quadratic form is positive definite if and only if all the characteristic numbers of its matrix are positive, and negative definite if and only if they are negative.

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Problem 32.30

Let all the characteristic numbers of a real symmetric matrix $A$ belong to the interval $\left[a, b\right]$ . Prove that the quadratic form with matrix $A - \lambda E$ is positive definite for $\lambda < a$ and negative definite for $\lambda > b$ .

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Problem 32.31

Prove that a quadratic form is positive definite if and only if all the coefficients of the characteristic polynomial of its matrix are nonzero and the signs of these coefficients alternate, with the constant term positive.

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Problem 32.32

(1)(2)

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(1)

Prove that the linear transformation $\varphi$ of a Euclidean space associated with a symmetric bilinear function $\mathrm{b}(x, y)$ given on it is self-adjoint.

(2)

Let in some basis $e$ a bilinear function $\mathrm{b}(x, y)$ have symmetric matrix $B$ , and let the Gram matrix of the basis $e$ equal $\Gamma$ . Find the matrix of the transformation associated with the function $\mathrm{b}(x, y)$ .

Problem 32.33

(1)(2)(3)(4)(5)(6)

In a basis $e$ of a Euclidean space, a quadratic form is given. Find, in the same basis, the matrix of the transformation associated with it, if the Gram matrix of the basis $e$ equals $\Gamma$ :

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(1)

$4x_{1}^{2} + 16x_{1} x_{2} + 6x_{2}^{2}, \quad \Gamma = \mathbf{A}_{56}$ . $\mathbf{A}_{56} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 3 \end{smallmatrix}\right]$ .

(2)

$4x_{1}^{2} - 6x_{1} x_{2} - 5x_{2}^{2}, \quad \Gamma = \mathbf{A}_{55}$ . $\mathbf{A}_{55} = \left[\begin{smallmatrix} 2 & 3 \\ 3 & 5 \end{smallmatrix}\right]$ .

(3)

$2x_{1} x_{2} - x_{2}^{2}, \quad \Gamma = \mathbf{A}_{9}$ . $\mathbf{A}_{9} = \left[\begin{smallmatrix} 9 & -5 \\ -5 & 3 \end{smallmatrix}\right]$ .

(4)

$2x_{1}^{2} + 4x_{1} x_{2} - 2x_{1} x_{3} - x_{2}^{2} + 4x_{2} x_{3} + 2x_{3}^{2}, \quad \Gamma = \mathbf{A}_{207}$ . $\mathbf{A}_{207} = \left[\begin{smallmatrix} 1 & 2 & 0 \\ 2 & 5 & -2 \\ 0 & -2 & 5 \end{smallmatrix}\right]$ .

(5)

$5x_{1}^{2} + x_{2}^{2} + 4x_{3}^{2} + 2x_{1} x_{2} + 4x_{1} x_{3} + 4x_{2} x_{3}, \quad \Gamma = \mathbf{A}_{308}$ . $\mathbf{A}_{308} = \left[\begin{smallmatrix} 5 & -1 & 2 \\ -1 & 1 & 0 \\ 2 & 0 & 4 \end{smallmatrix}\right]$ .

(6)

$x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + 2x_{1} x_{2} + 2x_{1} x_{3} + 2x_{1} x_{4} - 2x_{2} x_{3} - - 2x_{2} x_{4} - 2x_{3} x_{4}, \quad \Gamma = \mathbf{A}_{471}$ . $\mathbf{A}_{471} = \left[\begin{smallmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & -1 \\ 0 & -1 & 2 & 0 \\ 0 & -1 & 0 & 2 \end{smallmatrix}\right]$ .

Problem 32.34

Let in some basis $e$ of an $n$ -dimensional Euclidean space with Gram matrix $\Gamma$ a quadratic function $\mathrm{k}(x)$ have matrix $B$ . Prove that an orthonormal basis in which $\mathrm{k}(x)$ is diagonal, and its diagonal coefficients in this basis, are found as the solution of the generalized eigenvalue-eigenvector problem: $B\xi = \lambda \Gamma \xi$ ( $\xi \in \mathbb {R}^{n}$ ).

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Problem 32.35

Let $\mathcal{M} - r$ -dimensional linear subspace of an $n$ -dimensional Euclidean space. The function $\mathrm{k}(x)$ equals the square of the length of the orthogonal projection of the vector $x$ onto the subspace $\mathcal{M}$ . Prove that the function $\mathrm{k}(x)$ is quadratic. Find the diagonal form that this function has in some orthonormal basis.

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Problem 32.36

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)

Verify that at least one of the two given quadratic forms is sign-definite. Find a change of coordinates reducing these two forms simultaneously to diagonal form, and write this diagonal form for both forms.

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(1)

$\mathrm{f} = x_{1}^{2} + 2x_{1} x_{2} + 3x_{2}^{2}$ , $\mathrm{g} = 4x_{1}^{2} + 16x_{1} x_{2} + 6x_{2}^{2}$

(2)

$\mathrm{f} = 2x_{1}^{2} - 3x_{1} x_{2} - \frac{5}{2} x_{2}^{2}, \quad \mathrm{~ g} = 2x_{1}^{2} + 6x_{1} x_{2} + 5x_{2}^{2}$

(3)

$\mathrm{f} = 11x_{1}^{2} - 6x_{1} x_{2} + x_{2}^{2}, \quad \mathrm{~ g} = 13x_{1}^{2} - 10x_{1} x_{2} + 3x_{2}^{2}$

(4)

$\mathrm{f} = 9x_{1}^{2} - 10x_{1} x_{2} + 3x_{2}^{2}, \quad \mathrm{~ g} = 2x_{1} x_{2} - x_{2}^{2}$

(5)

$\mathrm{f} = x_{1}^{2} - 2x_{1} x_{2} + x_{2}^{2}, \quad \mathrm{~ g} = 17x_{1}^{2} + 8x_{1} x_{2} + x_{2}^{2}$

(6)

$\mathrm{f} = x_{1}^{2} + 2x_{1} x_{2} + 5x_{2}^{2}, \quad \mathrm{~ g} = 2x_{1} x_{2} - \frac{7}{2} x_{1}^{2} - x_{2}^{2}$

(7)

$\mathrm{f} = (1 + 4\sqrt{6}) x_{1}^{2} + 2\sqrt{6} x_{1} x_{2}, \quad \mathrm{~ g} = 5x_{1}^{2} + 4x_{1} x_{2} + x_{2}^{2}$

(8)

$\mathrm{f} = \left(1 + 2m\sqrt{a^{2} + a}\right) x_{1}^{2} + 2\sqrt{a^{2} + a} x_{1} x_{2}, \quad \mathrm{~ g} = \left(1 + m^{2}\right) x_{1}^{2} + 2mx_{1} x_{2} + x_{2}^{2}$ , where $m$ and $a$ are real parameters, $a^{2} + a \geqslant 0$

(9)

$\mathrm{f} = 5x_{1}^{2} + 2x_{1} x_{2} + 4x_{1} x_{3} + x_{2}^{2} + 4x_{2} x_{3} + 4x_{3}^{2}, \quad \mathrm{~ g} = 5x_{1}^{2} - - 2x_{1} x_{2} + 4x_{1} x_{3} + x_{2}^{2} + 2x_{3}^{2}$

(10)

$\mathrm{f} = 15x_{2}^{2} - 4x_{3}^{2} - 10x_{1} x_{2} - 8x_{1} x_{3} + 22x_{2} x_{3}, \quad \mathrm{~ g} = x_{1}^{2} - - 2x_{1} x_{3} + 4x_{2}^{2} + 4x_{2} x_{3} + 5x_{3}^{2}$

(11)

(р) $\mathrm{f} = 6x_{1}^{2} + 6x_{1} x_{3} + x_{2}^{2} - 6x_{2} x_{3} + 6x_{3}^{2}, \mathrm{~ g} = 2x_{1}^{2} + 2x_{1} x_{3} + x_{2}^{2} - 2x_{2} x_{3} + 2x_{3}^{2}$

(12)

$\mathrm{f} = 2x_{1}^{2} - 2x_{1} x_{2} - 2x_{1} x_{3} + x_{2}^{2} + 2x_{3}^{2}, \quad \mathrm{~ g} = 9x_{1}^{2} - 12x_{1} x_{2} - 24x_{1} x_{3} + 4x_{2}^{2} + 16x_{2} x_{3} + 16x_{3}^{2}$

(13)

$\mathrm{f} = x_{1}^{2} + 7x_{2}^{2} + 16x_{3}^{2} + 19x_{4}^{2} - 4x_{1} x_{2} + 10x_{1} x_{3} - - 10x_{1} x_{4} - 26x_{2} x_{3} + 8x_{2} x_{4} - 2x_{3} x_{4}, \quad \mathrm{~ g} = - x_{1}^{2} + 2x_{1} x_{2} - 2x_{2}^{2} + 4x_{2} x_{3} - 5x_{3}^{2} + 6x_{3} x_{4} - 10x_{4}^{2}$

(14)

$\mathrm{f} = x_{1}^{2} - 4x_{2} x_{3} + 4x_{3}^{2} - 4x_{3} x_{4} + 4x_{4}^{2}, \quad \mathrm{~ g} = x_{1}^{2} - 2x_{1} x_{2} + 2x_{2}^{2} - 2x_{2} x_{3} + 2x_{3}^{2} - 2x_{3} x_{4} + 2x_{4}^{2}$ .

Problem 32.37

Prove that if among the linear combinations of two quadratic functions there is a positive definite one, then these two functions are simultaneously reducible to diagonal form. Show by an example that this condition is not necessary.

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Problem 32.38

(1)(2)

In some basis $e$ of an $n$ -dimensional vector space, the quadratic functions $\mathrm{f}$ and $\mathrm{g}$ have matrices $F$ and $G$ , respectively. Let, in the basis $\mathbf{e}^{\prime }$ given by the transition matrix $S$ from the basis $e$ , the function $g$ have canonical form $\sum_{i = 1}^{n}{\xi_{i}^{\prime }}^{2}$ , and the function $\mathrm{f}$ have diagonal form $\sum_{i = 1}^{n} \lambda_{i}{\xi_{i}}^{2}$ . Prove that:

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(1)

$\operatorname {det}(F - \lambda G) = (\operatorname {det} S)^{2}\left(\lambda_{1} - \lambda \right) \ldots \left(\lambda_{n} - \lambda \right)$

(2)

the vectors of the basis $\mathbf{e}^{\prime }$ are found from the system of equations $(F - - \lambda G) \xi =$ o for each root $\lambda$ of the equation $\operatorname {det}(F - \lambda G) = 0$ .

Problem 32.39

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

Without finding the change of coordinates that reduces a positive definite quadratic form $g$ to canonical form, and a quadratic form $f$ to diagonal form, find this diagonal form of $\mathrm{f}$ .

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(1)

$\mathrm{f} = x_{1}^{2} + 2x_{1} x_{2} + x_{2}^{2}, \quad \mathrm{~ g} = 10x_{1}^{2} + 6x_{1} x_{2} + x_{2}^{2}$

(2)

$\mathrm{f} = 89x_{1}^{2} - 42x_{1} x_{2} + 5x_{2}^{2}, \quad \mathrm{~ g} = 41x_{1}^{2} - 18x_{1} x_{2} + 2x_{2}^{2}$

(3)

$\mathrm{f} = 7x_{1} x_{2} + 31x_{2}^{2}, \quad \mathrm{~ g} = x_{1}^{2} + 2x_{1} x_{2} + 2x_{2}^{2}$

(4)

$\mathrm{f} = 8x_{1}^{2} - 5x_{1} x_{2} + \frac{1}{2} x_{2}^{2}, \quad \mathrm{~ g} = x_{1}^{2} - x_{1} x_{2} + \frac{1}{2} x_{2}^{2}$ .

Subsection 32.3Bilinear and quadratic functions in complex space (32.40-32.47)[8/75%]

Problem 32.40

(1)(2)

Show that:

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(1)

if $\mathrm{b}(\mathbf{x}, \mathbf{y})$ is a bilinear function in an $n$ -dimensional complex arithmetic space, then the function $\mathrm{h}(\mathbf{x}, \mathbf{y}) = \mathrm{b}(\mathbf{x}, \overline{\mathbf{y}})$ is Hermitian bilinear

(2)

if b (x, y) is a Hermitian bilinear function in the space $\mathcal{C}_{n}$ , then $\mathrm{b}(\mathbf{x}, \mathbf{y}) = \mathrm{h}(\mathbf{x}, \overline{\mathbf{y}})$ is a bilinear function.

Problem 32.41

(1)(2)(3)(4)(5)(6)(7)

Reduce the following quadratic forms to canonical form:

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(1)

$4x_{1}^{2} - 12ix_{1} x_{2} - 9x_{2}^{2}$

(2)

$9x_{1}^{2} + 24(i + 1) x_{1} x_{2} + 16x_{2}^{2}$

(3)

$x_{1} x_{2}$

(4)

$\varepsilon^{2} x_{1}^{2} - \varepsilon x_{1} x_{2} + x_{2}^{2}, \varepsilon = e^{2\pi i / 3}$

(5)

$(1 + i) x_{1}^{2} + (2 + 2i) x_{1} x_{3} + ix_{2}^{2} + 3x_{3}^{2}$

(6)

$x_{1}^{2} + (2 - 2i) x_{1} x_{2} + 2x_{1} x_{3} + 2ix_{2}^{2} + (2 + 2i) x_{2} x_{3} + (1 + i) x_{3}^{2}$

(7)

$- x_{1}^{2} - 4ix_{1} x_{2} - (2 - 2i) x_{1} x_{3} + 4x_{2}^{2} - (4 + 4i) x_{2} x_{3} + 2ix_{3}^{2}$ .

Problem 32.42

(1)(2)(3)(4)(5)(6)(7)(8)(9)

Form the matrices of the given Hermitian bilinear forms in an $n$ -dimensional space:

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(1)

$- ix_{1} \bar{y}_{1} \quad (n = 1)$

(2)

$- ix_{1} \bar{y}_{1} \quad (n = 2)$

(3)

$3x_{1} \bar{y}_{1} + 4ix_{1} \bar{y}_{2} - 5x_{2} \bar{y}_{1} + ix_{2} \bar{y}_{2} \quad (n = 2)$

(4)

$- 3ix_{1} \bar{y}_{1} + 2x_{1} \bar{y}_{2} + 2x_{2} \bar{y}_{1} + (1 - i) x_{2} \bar{y}_{2} \quad (n = 2)$

(5)

$(1 + i) x_{1} \bar{y}_{2} + (1 + i) x_{2} \bar{y}_{1} - 5x_{2} \bar{y}_{2} \quad (n = 2)$

(6)

$(1 + i) x_{1} \bar{y}_{2} + (1 - i) x_{2} \bar{y}_{1} - 5x_{2} \bar{y}_{2} \quad (n = 2)$

(7)

$x_{1} \bar{y}_{1} - 3x_{2} \bar{y}_{2} + (2 + i) x_{3} \bar{y}_{3} - ix_{1} \bar{y}_{2} + (4 + i) x_{3} \bar{y}_{1} \quad (n = 3)$

(8)

$2x_{1} \bar{y}_{1} - 6x_{2} \bar{y}_{2} + (1 + 3\sqrt{2}) x_{3} \bar{y}_{3} + 3x_{1} \bar{y}_{2} + 3x_{2} \bar{y}_{1} + (2 - 5i) x_{1} \bar{y}_{3} + (2 + 5i) x_{3} \bar{y}_{1} + 4ix_{2} \bar{y}_{3} - 4ix_{3} \bar{y}_{2} \quad (n = 3)$

(9)

$\sum_{i = 1}^{n} x_{i} \bar{y}_{i}$

Problem 32.43

Which of the Hermitian bilinear forms in problem 32. 42 are symmetric? Write the corresponding Hermitian quadratic forms.

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Problem 32.44

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

Write the Hermitian quadratic form having the given matrix:

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(1)

$\mathbf{A}_{47}$ . $\mathbf{A}_{47} = \left[\begin{smallmatrix} 5 & -2 \\ -2 & 8 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{79}$ (при $\varepsilon = e^{2\pii / 3}$ ). $\mathbf{A}_{79} = \left[\begin{smallmatrix} 0 & \varepsilon \\ \varepsilon^{-1} & 0 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{280}$ . $\mathbf{A}_{280} = \left[\begin{smallmatrix} 0 & 2 & 1 \\ 2 & 8 & 2 \\ 1 & 2 & 0 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{491}$ . $\mathbf{A}_{491} = \left[\begin{smallmatrix} 3 & 1 & 0 & 2 i \\ 1 & 3 & -2 i & 0 \\ 0 & 2 i & 1 & 1 \\ -2 i & 0 & 1 & 1 \end{smallmatrix}\right]$ .

Problem 32.45

(1)(2)(3)(4)(5)(6)

A Hermitian quadratic form is written in an orthonormal basis of an $n$ -dimensional unitary space. Find an orthonormal basis in which the given Hermitian quadratic form has diagonal form, and write this diagonal form:

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(1)

$2\left|x_{1}\right|^{2} + ix_{1} \bar{x}_{2} - ix_{2} \bar{x}_{1} + 2\left|x_{2}\right|^{2} \quad (n = 2)$

(2)

$\left|x_{1}\right|^{2} + (3 - 4i) x_{1} \bar{x}_{2} + (3 + 4i) x_{2} \bar{x}_{1} + \left|x_{2}\right|^{2} \quad (n = 2)$

(3)

$\left|x_{1}\right|^{2} + \varepsilon x_{1} \bar{x}_{2} + \bar{\varepsilon } x_{2} \bar{x}_{1} + \left|x_{2}\right|^{2} \quad \left(\varepsilon = e^{2\pii / 3}\right) \quad (n = 2)$

(4)

$3\left|x_{1}\right|^{2} + 3\left|x_{2}\right|^{2} - 5\left|x_{3}\right|^{2} - ix_{1} \bar{x}_{2} + ix_{2} \bar{x}_{1} \quad (n = 3)$

(5)

$\left|x_{1}\right|^{2} + \left|x_{2}\right|^{2} + \left|x_{3}\right|^{2} + x_{1} \bar{x}_{2} + x_{2} \bar{x}_{1} + ix_{1} \bar{x}_{3} - ix_{3} \bar{x}_{1} + ix_{2} \bar{x}_{3} - - ix_{3} \bar{x}_{2} \quad (n = 3)$

(6)

$12\left|x_{1}\right|^{2} - (1 + i) x_{1} \bar{x}_{2} - (1 - i) x_{2} \bar{x}_{1} + 2x_{1} \bar{x}_{3} + 2x_{3} \bar{x}_{1} + (3 + 3i) x_{1} \bar{x}_{4} + (3 - 3i) x_{4} \bar{x}_{1} + 12\left|x_{2}\right|^{2} + (1 - i) x_{2} \bar{x}_{3} + (1 + i) x_{3} \bar{x}_{2} - - 2x_{2} \bar{x}_{4} - 2x_{4} \bar{x}_{2} + 8\left|x_{3}\right|^{2} - (1 + i) x_{3} \bar{x}_{4} - (1 - i) x_{4} \bar{x}_{3} + 8\left|x_{4}\right|^{2} (n = 4)$ .

Problem 32.46

Recover the symmetric Hermitian bilinear function $\mathrm{h}(x, y)$ from the Hermitian quadratic function $\mathrm{k}(x) = \mathrm{h}(x, x)$ .

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Problem 32.47

Prove that in the vector space of complex matrices of order $n$ the function $\mathrm{k}(X) = \operatorname {tr}\left(X\cdot \bar{X}^{T}\right)$ is a positive definite Hermitian quadratic function.

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