All textbooks/Collection of problems in analytical geometry and linear algebra/10 · Euclidean and unitary spaces

Section 25

Scalar product. Gram matrix

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25.1 · Definition of Euclidean space (25.1-25.19)

25.2 · Scalar product in coordinates (25.20-25.44)

25.3 · Orthogonal matrices (25.45-25.58)

Subsection 25.1Definition of Euclidean space (25.1-25.19)[19/47%]

Problem 25.1

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

Let $\mathbf{n}$ be a fixed nonzero vector in geometric space. To an arbitrary pair of vectors $\mathbf{x}, \mathbf{y}$ let us assign:

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(1)

the mixed product $(\mathbf{n}, \mathbf{x}, \mathbf{y})$

(2)

the inner product $(\mathbf{x} + \mathbf{n}, \mathbf{y} + \mathbf{n})$

(3)

$(\mathbf{n}, \mathbf{x})(\mathbf{n}, \mathbf{y})$ ;

(4)

$\left|\mathbf{n}\right|(\mathbf{x}, \mathbf{y})$ ;

(5)

$\left|\mathbf{x}\right|\left|\mathbf{y}\right|$ .

Can such a function be taken as an inner product?

Problem 25.2

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

Can the inner product in a real $n$ -dimensional vector space be given by the following function of the coordinates of the vectors:

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(1)

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

(1) $n = 2$ ;

(2) $n = 3$

(2)

$x_{1} y_{1} + x_{2}, \quad n = 2$

(3)

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

(4)

$2x_{1} y_{1} + x_{1} y_{2} + x_{2} y_{1} + 2x_{2} y_{2}, \quad n = 2$

(5)

$x_{1} y_{1} + 2x_{1} y_{2} + 2x_{2} y_{1} + x_{2} y_{2} + x_{3} y_{3}, \quad n = 3$ .

Problem 25.3

An ellipse with semi-axes 2 and 1 is drawn in the plane. Let a vector $\mathbf{x}$ be given. Consider the vector $\mathbf{x}_{0}$ , codirectional with $\mathbf{x}$ , whose start is the center of the ellipse and whose end lies on the ellipse. Set $\varphi (\mathbf{x}) = \left|\mathbf{x}\right| / \left|\mathbf{x}_{0}\right|$ , and $\varphi (\mathbf{o}) = 0$ . Now to an arbitrary pair of vectors one can assign the number

F(\mathbf{x}, \mathbf{y}) = \varphi ^{2}(\mathbf{x} + \mathbf{y}) - \varphi ^{2}(\mathbf{x} - \mathbf{y}).

Prove that this defines an inner product. Find its expression in terms of the coordinates of the vectors in the canonical coordinate system of the ellipse.

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Problem 25.4

(1)(2)(3)

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(1)

Prove that the function $F(X, Y) = \operatorname {tr} X^{T} Y$ is written in terms of the entries of the matrices by formula (1) of the introduction, and can be taken as an inner product in the space of real matrices of size $m \times n$ .

(2)

The Euclidean norm of a matrix is called its length with respect to this inner product. Prove that the Euclidean norm equals the square root of the sum of the squares of all entries of the matrix.

(3)

Find the lengths of the vectors of the standard basis and the angles between them with respect to this inner product.

Problem 25.5

(1)(2)(3)

Consider the space of square matrices of order $n$ , and to each pair of matrices assign a number:

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(1)

$F(X, Y) = \operatorname {tr} XY$

(2)

$F(X, Y) = \operatorname {tr} X\operatorname {tr} Y$

(3)

$F(X, Y) = \operatorname {det} XY$ .

Can such a function be taken as an inner product?

Problem 25.6

Let $P$ be a fixed square matrix of order $m$ . Under what conditions on this matrix is the function $F(X, Y) = \operatorname {tr} X^{T} PY$ an inner product in the vector space of matrices of size $m \times n$ ?

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Problem 25.7

In the vector space of functions continuous on the segment $\left[ - 1, 1\right]$ , to the functions $f$ and $g$ let us assign the number

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

Prove that this defines an inner product.

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Problem 25.8

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

In the vector space of polynomials of degree at most $n$ , to two polynomials $p$ and $q$ a number $F(p, q)$ is assigned. Prove that this defines an inner product:

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(1)

$F(p, q) = \int_{ - 1}^{1} p(t) q(t) dt$

(2)

$F$ is the sum of products of the derivatives of order $k$ , evaluated at the point $t_{0}$ ,

F(p, q) = \sum _{k = 0}^{n} p^{(k)}\left(t_{0}\right) q^{(k)}\left(t_{0}\right)

(3)

$F$ is the sum of products of the coefficients of equal powers,

F(p, q) = \sum _{k = 0}^{n} \alpha _{k} \beta _{k}

(4)

$F$ is the sum of products of the values of $p$ and $q$ at $m > n$ distinct points,

F(p, q) = \sum _{k = 1}^{m} p\left(t_{k}\right) q\left(t_{k}\right)

. (Verify that the requirement $m > n$ is necessary.)

Problem 25.9

Prove that in the Euclidean space of problem $25. 8, 1)$ the Legendre polynomials

P_{0}(t) = 1, \quad P_{k}(t) = \frac{1}{2^{k} k!} \frac{d^{k}}{dt^{k}}\left(t^{2} - 1\right)^{k} \quad (k = 1, \ldots , n)

form an orthogonal basis. Find the lengths (norms) of these polynomials.

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Problem 25.10

Let $\mathbf{e}$ be a basis in a vector space $\mathcal{E}$ . Prove that there exists in $\mathcal{E}$ one and only one inner product with respect to which the basis $\mathbf{e}$ is orthonormal.

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Problem 25.11

Let two inner products $(x, y)_{1}$ and $(x, y)_{2}$ be given in a real vector space. Prove that for any positive numbers $\lambda$ and $\mu$ the function $(x, y) = \lambda (x, y)_{1} + \mu (x, y)_{2}$ is also an inner product.

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Problem 25.12

A vector space $\mathcal{L}$ is given as the direct sum of subspaces $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ , in which inner products $(x, y)_{1}$ and $(x, y)_{2}$ are given. Let $x = x_{1} + x_{2}$ and $y = y_{1} + y_{2}$ , where $x_{1}, y_{1} \in \mathcal{L}_{1}; x_{2}, y_{2} \in \mathcal{L}_{2}$ . Prove that the function $(x, y) = \left(x_{1}, y_{1}\right)_{1} + \left(x_{2}, y_{2}\right)_{2}$ is an inner product on $\mathcal{L}$ .

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Problem 25.13

(1)(2)

Let $\left|x\right|$ be the length of a vector $x$ in a Euclidean space. Prove that:

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(1)

$(x, y) = \frac{1}{4}\left(\left|x + y\right|^{2} - \left|x - y\right|^{2}\right)$

(2)

$(x, y) = \frac{1}{2}\left(\left|x + y\right|^{2} - \left|x\right|^{2} - \left|y\right|^{2}\right)$ .

Problem 25.14

Let two inner products $(x, y)_{1}$ and $(x, y)_{2}$ be given in a real vector space, and let every vector have the same length in each of them: $(x, x)_{1} = (x, x)_{2}$ . Prove that the inner products coincide.

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Problem 25.15

(1)(2)

In the space of polynomials of degree $\leqslant 3$ with the standard inner product, a triangle is given with sides $t, t^{3}$ and $t - t^{3}$ . Find

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the angles of the triangle

(2)

the lengths of its sides.

Problem 25.16

Prove that a triangle in a Euclidean space is right-angled if and only if the length of one of the sides equals the length of another side multiplied by the cosine of the angle between these sides.

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Problem 25.17

Prove that a median of a triangle in a Euclidean space is shorter than one of the sides between which it lies.

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Problem 25.18

Prove that the sum of the angles of an arbitrary triangle in a Euclidean space equals $\pi$ .

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Problem 25.19

Let $\mathcal{L}$ be a finite-dimensional vector space. Prove that choosing a specific isomorphism of $\mathcal{L}$ onto its dual $\mathcal{L}^{ * }$ is equivalent to giving an inner product in $\mathcal{L}$ .

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Subsection 25.2Scalar product in coordinates (25.20-25.44)[25/80%]

Problem 25.20

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

In the arithmetic space with the standard inner product, find the inner products of

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(1)

$\left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 1 \\ 2 \\ 5 \\ 8\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 4\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 1 \\ - 1 \\ 1 \\ 0\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} -1 \\ 2 \\ 1 \\ -4 \end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 4 \\ - 1 \\ - 2 \\ 1\end{smallmatrix}\right]$

(4)

$\mathbf{c}_{171}$ and $\mathbf{c}_{172}$ . $\mathbf{c}_{171} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \\ 0 \end{smallmatrix}\right], \mathbf{c}_{172} = \left[\begin{smallmatrix} 3 \\ 6 \\ 4 \\ 3 \end{smallmatrix}\right]$ .

(5)

$\mathbf{c}_{174}$ and $\mathbf{c}_{175}$ . $\mathbf{c}_{174} = \left[\begin{smallmatrix} -1 \\ 1 \\ 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{175} = \left[\begin{smallmatrix} 1 \\ 3 \\ 0 \\ 1 \end{smallmatrix}\right]$ .

Problem 25.21

Find the angle between an edge and a diagonal of an $n$ -dimensional cube.

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Problem 25.22

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

Find the lengths of the vectors and the cosines of the angles between them in problem 25.20:

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(1)

$\left[\begin{smallmatrix} 1 \\ 1 \\ 1 \\ 1\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 1 \\ 2 \\ 5 \\ 8\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} 1 \\ 2 \\ 1 \\ 4\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 1 \\ - 1 \\ 1 \\ 0\end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} -1 \\ 2 \\ 1 \\ -4 \end{smallmatrix}\right]$ and $\left[\begin{smallmatrix} 4 \\ - 1 \\ - 2 \\ 1\end{smallmatrix}\right]$

(4)

(5)

Problem 25.23

Let, in some basis, the square of the length of any vector $x$ equal the sum of the squares of its coordinates. Prove that the basis is orthonormal.

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Problem 25.24

Draw in the plane some basis with Gram matrix $\mathbf{A}_{19}$ . Describe the whole set of such bases. $\mathbf{A}_{19} = \left[\begin{smallmatrix} 1 & 1 \\ 1 & 2 \end{smallmatrix}\right]$ .

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Problem 25.25

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

Find the inner product of the vectors, if their coordinates in some basis and the Gram matrix $\Gamma$ of this basis are given:

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(1)

$\left[\begin{smallmatrix} 1 \\ 1 \\ 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 1 \\ 3 \\ 1\end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 1 & - 2 & 1 \\ - 2 & 5 & - 4 \\ 1 & - 4 & 6\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} -1 \\ -1 \\ 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 0 \\ 1 \\ 3\end{smallmatrix}\right], \quad \Gamma =\left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 5 & 8 \\ 3 & 8 & 14\end{smallmatrix}\right]$

(3)

$\mathbf{c}_{20}, \mathbf{c}_{23}, \Gamma = \mathbf{A}_{24}$ . $\mathbf{c}_{20} = \left[\begin{smallmatrix} 0 \\ -2 \end{smallmatrix}\right], \mathbf{c}_{23} = \left[\begin{smallmatrix} 2 \\ -1 \end{smallmatrix}\right], \mathbf{A}_{24} = \left[\begin{smallmatrix} 1 & -1 \\ -1 & 2 \end{smallmatrix}\right]$ .

(4)

$\mathbf{c}_{23}, \mathbf{c}_{36}, \Gamma = \mathbf{A}_{17}$ . $\mathbf{c}_{23} = \left[\begin{smallmatrix} 2 \\ -1 \end{smallmatrix}\right], \mathbf{c}_{36} = \left[\begin{smallmatrix} 3 \\ -3 \end{smallmatrix}\right], \mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right]$ .

(5)

$\left[\begin{smallmatrix} 1 \\ \cdots \\ 1\end{smallmatrix}\right], \mathbf{c}_{282}, \mathbf{A}_{630}$ . $\mathbf{c}_{282} = \left[\begin{smallmatrix} 1 \\ \mu \\ \mu^2 \\ \vdots \\ \mu^{n-1} \end{smallmatrix}\right], \mathbf{A}_{630} = \left[\begin{smallmatrix} 1 & 1 & 1 & \ldots & 1 & 1 \\ 1 & 2 & 1 & \ldots & 1 & 1 \\ 1 & 1 & 3 & \ldots & 1 & 1 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 1 & 1 & 1 & \ldots & 1 & n \end{smallmatrix}\right]$ .

Problem 25.26

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

Find the angles between vectors given by their coordinates:

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(1)

$\left[\begin{smallmatrix} 1 & 1 & 1 & 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 0 & 2 & 0 & 2\end{smallmatrix}\right]$ ; the basis is orthonormal

(2)

$\left[\begin{smallmatrix} 2 & 1 & \sqrt{2} & 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 0 & 2 & 0 & 2\end{smallmatrix}\right]$ ; the basis is orthonormal

(3)

$\left[\begin{smallmatrix} 5 & 0 & 2 & 1\end{smallmatrix}\right],\left[\begin{smallmatrix} 2 & 1 & 2 & 1\end{smallmatrix}\right]$ ; the basis is orthonormal

(4)

$\left[\begin{smallmatrix} 1 \\ 0\end{smallmatrix}\right],\left[\begin{smallmatrix} 0 \\ 1\end{smallmatrix}\right]$ ; the basis has Gram matrix $\left[\begin{smallmatrix} 1 & 1 \\ 1 & 4\end{smallmatrix}\right]$

(5)

$\mathbf{c}_{12}, \mathbf{c}_{33}$ ; the basis has Gram matrix $\mathbf{A}_{58}$ . $\mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{33} = \left[\begin{smallmatrix} 2 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{58} = \left[\begin{smallmatrix} 1 & -2 \\ -2 & 5 \end{smallmatrix}\right]$ .

(6)

$\mathbf{c}_{18}, \mathbf{c}_{12}$ ; the basis has Gram matrix $\mathbf{A}_{58}$ . $\mathbf{c}_{18} = \left[\begin{smallmatrix} 3 \\ 1 \end{smallmatrix}\right], \mathbf{c}_{12} = \left[\begin{smallmatrix} 1 \\ 1 \end{smallmatrix}\right], \mathbf{A}_{58} = \left[\begin{smallmatrix} 1 & -2 \\ -2 & 5 \end{smallmatrix}\right]$ .

Problem 25.27

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

Find the lengths of the vectors in problem 25.25:

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(1)

(2)

(3)

(4)

(5)

Problem 25.28

(1)(2)(3)

For which values of the parameters $\varepsilon , \alpha$ can the given matrices serve as Gram matrices in a Euclidean space:

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(1)

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

(2)

$\mathbf{A}_{77}$ . $\mathbf{A}_{77} = \left[\begin{smallmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{76}$ . $\mathbf{A}_{76} = \left[\begin{smallmatrix} \cos \alpha & \sin \alpha \\ \sin \alpha & -\cos \alpha \end{smallmatrix}\right]$ .

Problem 25.29

Prove that in an $n$ -dimensional space a square matrix $\Gamma$ of order $n$ can serve as the Gram matrix of some basis if and only if there exists a square matrix $S$ with nonzero determinant such that $\Gamma = S^{T} S$ .

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Problem 25.30

Prove that in an $n$ -dimensional space a square matrix $\Gamma$ of order $n$ can serve as the Gram matrix of some basis if and only if it is positive definite, that is, $\Gamma = \Gamma^{T}$ and $\xi^{T} \Gamma \xi > 0$ for any column $\xi \neq \mathbf{o}$ .

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Problem 25.31

(1)(2)(3)

Let $\Gamma$ be the Gram matrix of some basis e. Prove that the following matrices are also Gram matrices of some bases:

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(1)

$\Gamma^{ - 1}$

(2)

$\Gamma^{2}$

(3)

$\Gamma^{k}$ , where $k$ is an integer.

Problem 25.32

Let $\Gamma_{1}$ and $\Gamma_{2}$ be the Gram matrices of bases $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ . Prove that for any positive coefficients the matrix $\Gamma = \alpha_{1} \Gamma_{1} + \alpha_{2} \Gamma_{2}$ is also the Gram matrix of some basis.

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Problem 25.33

In the Gram matrix of some basis all elements are equal to either 0, or 1, or - 1. Prove that the basis is orthonormal.

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Problem 25.34

Prove that the largest element in absolute value of the Gram matrix lies on the main diagonal.

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Problem 25.35

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

Can the third row of the Gram matrix of some basis in a four-dimensional space be the row:

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(1)

$\left[\begin{smallmatrix} 1 & 1 & 1 & 1\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} -1 & -1 & -1 & -1 \end{smallmatrix}\right]$

(3)

$\left[\begin{smallmatrix} 1 & 0 & 1 & 0\end{smallmatrix}\right]$

(4)

$\left[\begin{smallmatrix} 0 & 1 & 0 & 1\end{smallmatrix}\right]$

(5)

$\left[\begin{smallmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{4} & \frac{1}{2} \end{smallmatrix}\right]$ .

Problem 25.36

(1)(2)

Find the Gram matrix of the standard basis of the space of square matrices of order two¹ with the scalar product defined in problem 25.6, if $P$ equals:

Footnotes

See the introduction to 15. ↩

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(1)

$\mathbf{A}_{17}$ . $\mathbf{A}_{17} = \left[\begin{smallmatrix} 3 & 5 \\ 5 & 9 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{24}$ . $\mathbf{A}_{24} = \left[\begin{smallmatrix} 1 & -1 \\ -1 & 2 \end{smallmatrix}\right]$ .

Problem 25.37

In the space of polynomials of degree at most two with the standard scalar product, find the Gram matrix of the standard basis.

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Problem 25.38

In the space of square matrices of order $n$ with the scalar product of problem 25. 4, find the Gram matrix of the standard basis.

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Problem 25.39

The Gram matrix of a basis e is given. Find the Gram matrix of its biorthogonal basis $\mathbf{e}^{ * }$ .

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Problem 25.40

Let any two distinct vectors from the system $x_{1}, \ldots , x_{n}$ form an angle of $\pi / 3$ . Prove that these vectors are linearly independent.

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Problem 25.41

(1)(2)

Two systems of vectors $x_{1}, \ldots , x_{p}$ and $y_{1}, \ldots , y_{p}$ are given, and from the scalar products $c_{ij} = \left(x_{i}, y_{j}\right)$ a matrix $C$ is formed.

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(1)

Prove that $\operatorname {det} C = 0$ if at least one of the systems is linearly dependent.

(2)

Is the converse true?

Problem 25.42

Two ordered systems of vectors $e_{1}, \ldots , e_{k}$ and $f_{1}, \ldots , f_{k}$ in a Euclidean space are called biorthogonal if $\left(e_{i}, f_{j}\right) = 0$ for $i \neq j$ , and $\left(e_{i}, f_{i}\right) = 1$ for all $i$ . Prove that each of the two biorthogonal systems is linearly independent.

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Problem 25.43

For a system of vectors $x_{1}, \ldots , x_{p}$ of a Euclidean space, a matrix $C$ is formed with elements $c_{ij} = \left(x_{i}, x_{j}\right)$ . Let $\operatorname {Rg} C = k$ and let the order- $k$ minor in the upper left corner be a basis minor. Indicate some maximal linearly independent subsystem of the given system of vectors.

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Problem 25.44

Using the properties of the matrix formed from all possible scalar products, prove that for any matrix $A$ we have $\operatorname {Rg} A^{T} A = \operatorname {Rg} A$ .

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Subsection 25.3Orthogonal matrices (25.45-25.58)[14/100%]

Problem 25.45

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

Which of the following matrices are orthogonal:

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(1)

$\mathbf{A}_{32}$ . $\mathbf{A}_{32} = \left[\begin{smallmatrix} 1 & 3 \\ 3 & 10 \end{smallmatrix}\right]$ .

(2)

$\mathbf{A}_{22}$ . $\mathbf{A}_{22} = \left[\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix}\right]$ .

(3)

$\mathbf{A}_{16}$ . $\mathbf{A}_{16} = \left[\begin{smallmatrix} 1 & -1 \\ 1 & 1 \end{smallmatrix}\right]$ .

(4)

$\mathbf{A}_{15}$ . $\mathbf{A}_{15} = \left[\begin{smallmatrix} -1 & 0 \\ 0 & 1 \end{smallmatrix}\right]$ .

(5)

$\mathbf{A}_{20}$ . $\mathbf{A}_{20} = \left[\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right]$ .

(6)

$\mathbf{A}_{64}$ . $\mathbf{A}_{64} = \left[\begin{smallmatrix} 3/\sqrt{10} & 1/\sqrt{10} \\ 1/\sqrt{10} & -3/\sqrt{10} \end{smallmatrix}\right]$ .

(7)

$\mathbf{A}_{243}$ . $\mathbf{A}_{243} = \left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{smallmatrix}\right]$ .

(8)

$\mathbf{A}_{253}$ . $\mathbf{A}_{253} = \left[\begin{smallmatrix} 0 & 1 & 1 \\ -1 & 0 & -1 \\ -1 & 1 & 0 \end{smallmatrix}\right]$ .

(9)

$\mathbf{A}_{255}$ . $\mathbf{A}_{255} = \left[\begin{smallmatrix} 0 & 1 & 0 \\ -1 & 0 & 3 \\ 0 & -3 & 0 \end{smallmatrix}\right]$ .

(10)

$\mathbf{A}_{329}$ . $\mathbf{A}_{329} = \left[\begin{smallmatrix} 1/\sqrt{2} & 1/\sqrt{6} & 1/\sqrt{3} \\ 1/\sqrt{2} & -1/\sqrt{6} & -1/\sqrt{3} \\ 0 & 2/\sqrt{6} & -1/\sqrt{3} \end{smallmatrix}\right]$ .

(11)

$\mathbf{A}_{330}$ . $\mathbf{A}_{330} = \left[\begin{smallmatrix} 1/\sqrt{3} & -1/\sqrt{3} & 1/\sqrt{3} \\ 1/(3 - \sqrt{3}) & 1/(3 + \sqrt{3}) & -1/\sqrt{3} \\ 1/(3 + \sqrt{3}) & 1/(3 - \sqrt{3}) & 1/\sqrt{3} \end{smallmatrix}\right]$ .

(12)

$\mathbf{A}_{432}$ . $\mathbf{A}_{432} = \left[\begin{smallmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \end{smallmatrix}\right]$ .

(13)

$\mathbf{A}_{433}$ . $\mathbf{A}_{433} = \left[\begin{smallmatrix} 1 & -2 & 1 & 0 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 1 \end{smallmatrix}\right]$ .

(14)

$\mathbf{A}_{445}$ . $\mathbf{A}_{445} = \left[\begin{smallmatrix} 1/\sqrt{10} & 1/2 & 1/2 & 2/\sqrt{10} \\ 2/\sqrt{10} & 1/2 & -1/2 & -1/\sqrt{10} \\ 1/\sqrt{10} & -1/2 & -1/2 & 2/\sqrt{10} \\ 2/\sqrt{10} & -1/2 & 1/2 & -1/\sqrt{10} \end{smallmatrix}\right]$ .

(15)

$\mathbf{A}_{436}$ . $\mathbf{A}_{436} = \left[\begin{smallmatrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{smallmatrix}\right]$ .

Problem 25.46

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

Will an orthogonal matrix remain orthogonal if:

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(1)

its rows are permuted

(2)

its columns are permuted

(3)

the elements of rows with odd numbers are written in reverse order

(4)

it is transposed

(5)

it is rotated about the anti-diagonal

(6)

one of its rows is multiplied by a number

(7)

one of its rows is added to another.

Problem 25.47

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

Let $A$ and $B$ be orthogonal matrices of the same order. Are the following matrices orthogonal:

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(1)

$A + B$

(2)

$AB$

(3)

$AB^{T}$

(4)

$\alpha A$

(5)

$A^{k}, \quad k$ an integer.

Problem 25.48

Find all pairs of orthogonal matrices of order two whose sum is an orthogonal matrix.

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Problem 25.49

Under what condition does there exist, for an orthogonal matrix $A$ , a number $\alpha \neq 0$ such that the matrix $A + \alpha E$ is also orthogonal? Do such matrices, other than $E$ and $- E$ , exist for $n = 2, 3, 4$ ?

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Problem 25.50

(1)(2)

Can an orthogonal matrix of order four contain the row:

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(1)

$\left[\begin{smallmatrix} 1 & 0 & 1 & 0\end{smallmatrix}\right]$

(2)

$\left[\begin{smallmatrix} \frac{1}{2} & \frac{1}{2} & 0 & \frac{1}{\sqrt{2}} \end{smallmatrix}\right]?$

Problem 25.51

A row of length $n$ is given, the sum of the squares of whose elements equals 1. Does there exist an orthogonal matrix of order $n$ with such a row?

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Problem 25.52

Find all orthogonal matrices having first row $\left[\begin{smallmatrix} \frac{1}{2} & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{smallmatrix}\right]$ .

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Problem 25.53

(1)(2)

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(1)

Can all elements of an orthogonal matrix be positive?

(2)

Prove that an orthogonal matrix all of whose elements are nonnegative is obtained from the identity matrix by a permutation of columns.

Problem 25.54

Find all orthogonal matrices that are upper triangular.

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Problem 25.55

Under what condition will a submatrix of an orthogonal matrix also be orthogonal?

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Problem 25.56

Suppose that all elements of an orthogonal matrix of order $n$ are equal to one another in absolute value. What is the absolute value of the element equal to?

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Problem 25.57

Prove that the orthogonal matrices described in problem 25. 56 exist if $n = 2^{k}$ , $k$ a natural number.

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Problem 25.58

(1)(2)

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(1)

Two orthonormal bases $e_{1}, \ldots , e_{n}$ and $f_{1}, \ldots , f_{n}$ are given. Prove that the matrix of scalar products $\left(e_{i}, f_{j}\right)$ is orthogonal.

(2)

Two orthonormal systems of $k < n$ vectors each are given in an $n$ -dimensional Euclidean space. Under what condition is the matrix of pairwise scalar products of the vectors of these systems orthogonal?