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Chapter VIII

Bilinear and quadratic functions

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37 · General bilinear and sesquilinear functions

38 · Symmetric bilinear, Hermitian and quadratic functions

Section 37General bilinear and sesquilinear functions[46/63%]

Problem 37.1

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)(q)(r)(s)

Which of the following functions of two arguments are bilinear functions on the corresponding spaces:

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(a)

$f(x, y)={ }^{t} x \cdot y \quad \left(x, y \in F^{n}\right.$ are columns, $F$ a field);

(b)

$f(A, B)=\operatorname {tr}(A B) \quad \left(A, B \in \mathbf{M}_{n}(F), F\right.$ a field);

(c)

$f(A, B)=\operatorname {tr}(A B-B A)$ ;

(d)

$f(A, B)=\operatorname {det}(A B)$ ;

(e)

$f(A, B)=\operatorname {tr}(A+B)$ ;

(f)

$f(A, B)=\operatorname {tr}\left(A \cdot { }^{t} B\right)$ ;

(g)

$f(A, B)=\operatorname {tr}\left({ }^{t} A \cdot B\right)$ ;

(h)

$f(A, B)$ is the $(i, j)$ entry of the matrix $A B$ ;

(i)

$f(u, v)=\operatorname {Re}(u v) \quad (u, v \in \mathbb {C}, \mathbb {C}$ a vector space over $\mathbb {R})$ ;

(j)

$f(u, v)=\operatorname {Re}(u \bar{v})$ ;

(k)

$f(u, v)=\left|u v\right|$ ;

(l)

$f(u, v)=\operatorname {Im}(u \bar{v})$ ;

(m)

$f(u, v)=\int_{b}^{a} u v d t \quad (u, v$ are continuous functions of the argument $t$ on the interval $[a, b]$ );

(n)

$f(u, v)=\int_{b}^{a} u v^{\prime } d t \quad (u, v$ are differentiable functions on the interval $[a, b]$ , with $u(a)=u(b)=v(a)=v(b)=0)$ ;

(o)

$f(u, v)=\int_{b}^{a}(u+v)^{2} d t ;$

(p)

$f(u, v)=(u v)(\alpha ) \quad (u, v \in F[x], \quad \alpha \in F)$ ;

(q)

$f(u, v)=\frac{d}{d t}(u v)(a)$ ;

(r)

$f(u, v)=\left|u+v\right|^{2}-\left|u\right|^{2}-\left|v\right|^{2} \quad \left(u, v \in \mathbb {R}^{3}\right)$ ;

(s)

$f(u, v)=\varepsilon (u \times v)$ ( $\times$ is the cross product, $\varepsilon (x)$ is the sum of the coordinates of the vector $x$ in a given basis).

Problem 37.2

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)

In the finite-dimensional spaces of problem 37.1, choose a basis and find the matrices of the corresponding bilinear functions:

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(a)

$f(x, y)={ }^{t} x \cdot y \quad (x, y \in F^{n}$ are columns, $F$ a field);

(b)

$f(A, B)=\operatorname {tr}(A B) \quad (A, B \in \mathbf{M}_{n}(F)$ , $F$ a field);

(c)

$f(A, B)=\operatorname {tr}(A B-B A)$ ;

(d)

$f(A, B)=\operatorname {tr}\left(A \cdot { }^{t} B\right)$ ;

(e)

$f(A, B)=\operatorname {tr}\left({ }^{t} A \cdot B\right)$ ;

(f)

$f(A, B)$ is the $(i, j)$ entry of the matrix $A B$ (for fixed $i, j$ );

(g)

$f(u, v)=\operatorname {Re}(u v) \quad (u, v \in \mathbb {C}$ , $\mathbb {C}$ a vector space over $\mathbb {R})$ ;

(h)

$f(u, v)=\operatorname {Re}(u \bar{v})$ ;

(i)

$f(u, v)=\operatorname {Im}(u \bar{v})$ ;

(j)

$f(u, v)=\left|u+v\right|^{2}-\left|u\right|^{2}-\left|v\right|^{2} \quad (u, v \in \mathbb {R}^{3})$ ;

(k)

$f(u, v)=\varepsilon (u \times v)$ ( $\times$ is the cross product, $\varepsilon (x)$ is the sum of the coordinates of the vector $x$ in a given basis).

Problem 37.3

Let $F$ be a field and $F(x)$ the field of rational functions in the variable $x$ . Prove that the maps $x \mapsto \varepsilon x^{\tau }$ define automorphisms of order two in $F(x)$ , where $\varepsilon , \tau = \pm 1$ and $(\varepsilon , \tau ) \neq (1,1)$ .

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Problem 37.4

Let $p, q$ be distinct primes. Prove that the real numbers of the form $a+b \sqrt{p}+c \sqrt{q}+d \sqrt{p q}$ , where $a, b, c, d \in \mathbb {Q}$ , form a subfield $\mathbb {Q}(\sqrt{p}, \sqrt{q})$ of $\mathbb {R}$ . Check that the map

x \rightarrow \bar{x}=a-b \sqrt{p}+c \sqrt{q}-d \sqrt{p q}

is an automorphism of order two in this field.

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Problem 37.5

(a)(b)(c)(d)(e)(f)(g)(h)

Let $F$ be a field with an automorphism $a \rightarrow \bar{a}$ of order two. Which of the following functions of two arguments are sesquilinear functions on the corresponding vector spaces:

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(a)

$f(a, b)={ }^{t} a \cdot \bar{b} \quad \left(a, b \in F^{n}\right.$ are columns $)$ ;

(b)

$f(A, B)=\operatorname {tr}(A \bar{B}) \quad \left(A, B \in \mathbf{M}_{n}(F)\right)$ ;

(c)

$f(A, B)=\operatorname {det}(A \bar{B})$ ;

(d)

$f(A, B)=\operatorname {tr}\left(A \cdot { }^{t} \bar{B}\right)$ ;

(e)

$f(A, B)=\operatorname {tr}\left(A \cdot { }^{t} B\right)$ ;

(f)

$f(A, B)$ is the entry in position $(i, j)$ of the matrix $A \bar{B}$ ;

(g)

$f(A, B)$ is the entry in position $(i, j)$ of the matrix $A^{t} \bar{B}$ ;

(h)

$f(u, v)=\frac{d}{d x}(u \bar{v})(a),(u, v \in F[x], a \in F$ ; applying the automorphism to a polynomial means applying it to all of its coefficients).

Problem 37.6

(a)(b)

Find the matrix of the bilinear function $f$ in the new basis, given its matrix in the old basis and the transition formulas:

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(a)

$\left(\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right)$ , $e_{1}^{\prime }=e_{1}-e_{2}$ , $e_{2}^{\prime }=e_{1}+e_{3}$ , $e_{3}^{\prime }=e_{1}+e_{2}+e_{3} ;$

(b)

$\left(\begin{array}{rrr}0 & 2 & 1 \\ -2 & 2 & 0 \\ -1 & 0 & 3\end{array}\right), \quad \begin{aligned} & e_{1}^{\prime }=e_{1}+2 e_{2}-e_{3}, \\ & e_{2}^{\prime }=e_{2}-e_{3}, \\ & e_{3}^{\prime }=-e_{1}+e_{2}-3 e_{3}.\end{aligned}$

Problem 37.7

(a)(b)

Let a sesquilinear function $f$ on a two-dimensional complex space with basis $(e_{1}, e_{2})$ be given by the matrix $B$ . Find the matrix $B^{\prime }$ of the function $f$ in the basis $\left(e_{1}^{\prime }, e_{2}^{\prime }\right)$ , where:

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(a)

$B=\left(\begin{array}{cc}i+1 & -1 \\ 0 & -i\end{array}\right), \quad \begin{aligned} & e_{1}^{\prime }=e_{1}+i e_{2}, \\ & e_{2}^{\prime }=+i e_{1}+e_{2}\end{aligned}$

(b)

$B=\left(\begin{array}{rc}2 & -1+i \\ -i & 0\end{array}\right), \quad \begin{aligned} & e_{1}^{\prime }=2 e_{1}-i e_{2}, \\ & e_{2}^{\prime }=i e_{1}+e_{2}.\end{aligned}$

Problem 37.8

(a)(b)

Let a bilinear function $f$ be given in some basis by the matrix $F$ . Find $f(x, y)$ if:

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(a)

$F=\left(\begin{array}{rrr}1 & -1 & 1 \\ -2 & -1 & 3 \\ 0 & 4 & 5\end{array}\right), \quad \begin{aligned} & x=(1,0,3), \\ & y=(-1,2,-4) ;\end{aligned}$

(b)

$F=\left(\begin{array}{ccc}i & 1+i & 0 \\ -1+i & 0 & 2-i \\ 2+i & 3-i & -1\end{array}\right), \quad \begin{aligned} & x=(1+i, 1-i, 1), \\ & y=(-2+i,-i, 3+2 i).\end{aligned}$

Problem 37.9

(a)(b)

Find the value $f(x, y)$ of the sesquilinear function given in some basis of a complex space by the matrix $B$ , if:

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(a)

$B=\left(\begin{array}{rc}5 & 2 \\ -1 & i\end{array}\right), \quad \begin{aligned} & x=(i,-2), \\ & y=(1-i, 3+i) ;\end{aligned}$

(b)

$B=\left(\begin{array}{ll}-i & 1-2 i \\ -4 & 2+3 i\end{array}\right), \quad \begin{aligned} & x=(2, i+3), \\ & y=(-i, 6-2 i).\end{aligned}$

Problem 37.10

(a)(b)

Let $g$ be a bilinear function with matrix $G$ in some basis of the space $V$ , and let $\mathcal{A}$ be a linear operator on $V$ with matrix $A$ . Find, in this basis, the matrix of the bilinear function $f(u, v)=g(u$ , $\mathcal{A}(v)$ ), if:

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(a)

$G=\left(\begin{array}{rrr}1 & -1 & 0 \\ 2 & 0 & -2 \\ 3 & 4 & 5\end{array}\right), \quad A=\left(\begin{array}{rrr}-1 & 1 & 1 \\ -3 & -4 & 2 \\ 1 & -2 & -3\end{array}\right) ;$

(b)

$G=\left(\begin{array}{lll}0 & 1 & 2 \\ 4 & 0 & 3 \\ 5 & 6 & 0\end{array}\right), \quad A=\left(\begin{array}{rrr}1 & -4 & 3 \\ 4 & -1 & -2 \\ -3 & 2 & 1\end{array}\right)$ .

Problem 37.11

(a)(b)

Let $g$ be a sesquilinear function with matrix $G$ in some basis of a linear space $V$ , and let $\mathcal{A}$ be a linear operator on $V$ with matrix $A$ . Find, in this basis, the matrix of the sesquilinear function $f(u, v)=g(u, \mathcal{A}(v))$ , if:

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(a)

$G=\left(\begin{array}{cc}-5+i & 2 \\ -i & 4+i\end{array}\right), \quad A=\left(\begin{array}{cc}i & 0 \\ 2 & -i\end{array}\right) ;$

(b)

$G=\left(\begin{array}{cc}4-i & 2-i \\ 0 & 1+i\end{array}\right), \quad A=\left(\begin{array}{cc}-i+1 & 0 \\ 4 & 3+i\end{array}\right)$ .

Problem 37.12

(a)(b)

Find the left and right kernels of the bilinear function $f$ given in the basis $(e_{1}, e_{2}, e_{3})$ by the matrix:

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(a)

$\left(\begin{array}{lll}2 & -3 & 1 \\ 3 & -5 & 5 \\ 5 & -8 & 6\end{array}\right)$ ;

(b)

$\left(\begin{array}{lll}4 & 3 & 2 \\ 1 & 3 & 5 \\ 3 & 6 & 9\end{array}\right)$ .

Problem 37.13

(a)(b)

Let $F=\mathbb {Q}(\sqrt{3})$ and $\bar{x}=a-b \sqrt{3}$ if $x=a+b \sqrt{3}$ , where $a, b \in \mathbb {Q}$ . Find, in a two-dimensional vector space over $F$ , the left and right kernels of the sesquilinear function given in the basis $\left(e_{1}, e_{2}\right)$ by the matrix:

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(a)

$\left(\begin{array}{cc}1+\sqrt{3} & -1-\sqrt{3} \\ 1 & 2-\sqrt{3}\end{array}\right)$ ;

(b)

$\left(\begin{array}{cc}-3 & -2 \sqrt{3} \\ 2 \sqrt{3} & 4\end{array}\right)$ .

Problem 37.14

(a)(b)

Find the left and right kernels of the bilinear function $f(x, y)=(x, \mathcal{A}(y))$ , where $\mathcal{A}$ is a linear operator with matrix $A$ in an orthonormal basis $\left(e_{1}, e_{2}, e_{3}\right)$ of a Euclidean space:

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(a)

$A=\left(\begin{array}{rrr}5 & -6 & 1 \\ 3 & -5 & -2 \\ 2 & -1 & 3\end{array}\right)$ ;

(b)

$A=\left(\begin{array}{lll}2 & -1 & 3 \\ 3 & -2 & 2 \\ 5 & -4 & 0\end{array}\right)$ .

Problem 37.15

(a)(b)

Let $F, \bar{x}$ be as in 37.13. Find the left and right kernels of the sesquilinear function $f(x, y)=g(x, \mathcal{A}(y))$ , where $\mathcal{A}$ is a linear operator with matrix $A$ , and $g$ is the sesquilinear function on the two-dimensional space having the identity matrix in the basis $(e_{1}, e_{2})$ :

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(a)

$A=\left(\begin{array}{cc}1+\sqrt{3} & 1 \\ 0 & 0\end{array}\right)$ ;

(b)

$A=\left(\begin{array}{cc}0 & 0 \\ 2 \sqrt{3} & 0\end{array}\right)$ .

Problem 37.16

(a)(b)

Let $f$ be a bilinear function with matrix $F$ on a vector space $V$ , and let $U$ be a subspace of $V$ . Find the left and right orthogonal complements of $U$ with respect to $f$ (i.e., the maximal subspaces $U_{1}$ and $U_{2}$ such that $f\left(U_{1}, U\right)=f\left(U, U_{2}\right)=0$ ), if:

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(a)

$F=\left(\begin{array}{ccc}4 & 1 & 3 \\ 3 & 3 & 6 \\ 2 & 5 & 9\end{array}\right), \quad U=\langle (1,-1,0),(-2,3,1)\rangle$ ;

(b)

$F=\left(\begin{array}{lll}6 & -8 & 5 \\ 5 & -5 & 3 \\ 1 & -3 & 2\end{array}\right), \quad U=\langle (2,0,-3),(3,1,-5)\rangle$ .

Problem 37.17

(a)(b)

With $F, \bar{x}$ as in 37.13, let $f$ be a sesquilinear function with matrix $G$ on a vector space $V$ , and let $U$ be a subspace of $V$ . Find the left and right orthogonal complements of $U$ with respect to $f$ , if:

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(a)

$G=\left(\begin{array}{ccc}1+\sqrt{3} & 2 & -\sqrt{3} \\ 0 & 1 & 2 \\ 1-\sqrt{3} & 2 & 0\end{array}\right), \quad U=\langle (1,0, \sqrt{3}),(0,2,1)\rangle$ ;

(b)

$G=\left(\begin{array}{ccc}1 & -\sqrt{3} & 2 \\ 2 & 0 & 1 \\ 1 & 1-\sqrt{3} & \sqrt{3}\end{array}\right), \quad U=\langle (1,-\sqrt{3}, 2)\rangle$ .

Problem 37.18

(a)(b)(c)(d)

Let $F$ be a finite field of $q^{2}$ elements, with $\bar{x}=x^{q}$ for all $x \in F$ . Suppose $K$ is a finite field containing $F$ with $n=\operatorname {dim}_{F} K$ . Prove that:

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(a)

$x \rightarrow \bar{x}$ is an automorphism of order two in $F$ ;

(b)

the function

f(x, y)=x y^{q}+x^{q^{2}} y^{q^{3}}+\ldots +x^{q^{2 n-2}} y^{q^{2 n-1}}

is a sesquilinear function on $K$ regarded as a vector space over $F$ ;

(c)

the function $f(x, y)$ from b) is nondegenerate;

(d)

find the left and right orthogonal complements of $F$ in $K$ with respect to $f(x, y)$ .

Problem 37.19

(a)(b)(c)(d)(e)

Let $F=\mathbb {Q}\left[i\right], K=F[\sqrt{2}]$ . Consider $K$ as a vector space over $F$ . Prove that:

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(a)

$\operatorname {dim}_{F} K=2$ ;

(b)

the function $f\left(z_{1}+z_{2} \sqrt{2}, t_{1}+t_{2} \sqrt{2}\right)=z_{1} \bar{t}_{1}+2 z_{2} \bar{t}_{2}$ , where the bar denotes complex conjugation, is a sesquilinear function on $K$ regarded as a vector space over $F$ ;

(c)

find the matrix of $f$ in the basis $e_{1}=1, e_{2}=\sqrt{2}$ ;

(d)

prove that the function $f$ is nondegenerate;

(e)

find the left and right orthogonal complements of $F$ in $K$ with respect to $f$ .

Problem 37.20

(a)(b)(c)(d)(e)(f)

Let $F=\mathbb {C}(x)$ be the field of rational functions with the automorphism under which $x \rightarrow \varepsilon x^{\tau }$ , where $\varepsilon , \tau = \pm 1,(\varepsilon , \tau ) \neq (1,1)$ .

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(a)

Prove that the polynomial $y^{4}-x \in F[y]$ is irreducible over $F$ .

(b)

Prove that in the field $K=F[y] /\left(y^{4}-x\right)$ , regarded as a vector space over $F$ , the function $f(u, v)=u(x, y) v\left(i x^{\tau }, y\right)+u(x, i y) v\left(\varepsilon x^{\tau }, i y\right)+$

+u(x,-y) v\left(\varepsilon x^{\tau },-y\right)+u(x,-i y) v\left(\varepsilon x^{\tau },-i y\right)

is sesquilinear.

(c)

Find the matrix of $f(u, v)$ in the basis $1, y, y^{2}, y^{3}$ of the space $K$ over the field $F$ .

(d)

Prove that the function $f$ is nondegenerate.

(e)

Find the left and right orthogonal complements of the linear span $\langle 1, y\rangle$ in $K$ .

(f)

Find a basis $u_{0}, u_{1}, u_{2}, u_{3}$ of $K$ such that $f\left(u_{i}, y^{j}\right)=\delta_{i j}$ , where $i, j=0,1,2,3$ .

Problem 37.21

(a)(b)(c)

For which of the following elementary transformations of a basis does the matrix of a bilinear function change in the same way as the matrix of a linear operator:

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(a)

$\left(e_{1}, \ldots , e_{i}, \ldots , e_{n}\right) \rightarrow \left(e_{1}, \ldots , \lambda e_{i}, \ldots , e_{n}\right)$ ;

(b)

$\left(e_{1}, \ldots , e_{i}, \ldots , e_{n}\right) \rightarrow \left(e_{1}, \ldots , e_{i}+\lambda e_{j}, \ldots , e_{n}\right) \quad (j \neq i)$ ;

(c)

$\left(e_{1}, \ldots , e_{i}, \ldots , e_{j}, \ldots , e_{n}\right) \rightarrow \left(e_{1}, \ldots , e_{j}, \ldots , e_{i}, \ldots , e_{n}\right)$ ?

Problem 37.22

Find the relation between the matrices $A, B, G$ of the linear operators $\mathcal{A}, \mathcal{B}$ and the bilinear (sesquilinear) function $g$ in some basis of the space, and the matrix $F$ of the bilinear (sesquilinear) function

f(x, y)=g(\mathcal{A}(x), \mathcal{B}(y))

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Problem 37.23

Prove that every bilinear (sesquilinear) function $f$ of rank 1 can be represented as the product of two linear functions $p(x) q(y)$ (respectively, $p(x) q(\bar{y})$ ). To what simplest form can the matrix of $f$ be reduced by a change of basis?

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Problem 37.24

Let $\mathbf{e}=\left(e_{1}, \ldots , e_{n}\right), \mathbf{e}^{\prime }=\left(e_{1}^{\prime }, \ldots , e_{n}^{\prime }\right)$ be two bases of a space $V$ , let $C$ be the transition matrix from $\mathbf{e}$ to $\mathbf{e}^{\prime }$ , and let $f$ be a bilinear (sesquilinear) function on $V$ with matrices $F$ and $F^{\prime }$ in these bases. Find the relation between the matrices $F, F^{\prime }$ .

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Problem 37.25

Prove that the bilinear and sesquilinear functions $\operatorname {tr}(A B), \operatorname {tr}\left(A^{t} B\right), \operatorname {tr}(A \bar{B}), \operatorname {tr}\left(A^{t} \bar{B}\right)$ on the space $\mathbf{M}_{n}(K)$ are nondegenerate.

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Problem 37.26

Prove that the dimensions of the left and right kernels of a bilinear (sesquilinear) function coincide, but the kernels themselves need not coincide.

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Problem 37.27

Let $f$ be a nondegenerate bilinear (sesquilinear) function on a space $V$ . Prove that for any linear function $p$ there is a unique vector $v \in V$ such that $p(x)=f(x, v)$ for any $x \in V$ , and that the map $p \mapsto v$ is an isomorphism of the spaces $V^{*}$ and $V$ .

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Problem 37.28

(a)(b)(c)

Let $F$ be the matrix of a nondegenerate bilinear function $f$ on a real space of dimension $n$ .

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(a)

Prove that for odd $n$ the matrix $-F$ is not the matrix of $f$ in any basis of the space $V$ .

(b)

Is the assertion of a) true for even $n$ ?

(c)

Is the assertion of a) true for even $n$ when $F$ is a diagonal matrix?

Problem 37.29

(a)(b)(c)(d)

Suppose that for a nonzero bilinear (sesquilinear) function $f$ on a space $V$ there is a number $\varepsilon$ such that for any $x, y \in V$

f(y, x)=\varepsilon f(x, y)

(respectively, $\overline{f(y, x)}=\varepsilon f(x, y)$ ). Prove that:

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(a)

$\varepsilon$ equals 1 or $-1$ ;

(b)

if $U_{1}$ and $U_{2}$ are totally isotropic subspaces with respect to $f$ , having the same dimension, and $U_{1} \cap U_{2}^{\perp }=0$ , then the restriction of $f$ to their sum $U_{1}+U_{2}$ is a nondegenerate function;

(c)

if $W_{1}$ and $W_{2}$ are maximal totally isotropic subspaces with respect to $f$ and $W_{1} \cap W_{2}=0$ , then $\operatorname {dim} W_{1}=\operatorname {dim} W_{2}$ ;

(d)

if nondegenerate bilinear functions $f_{1}$ and $f_{2}$ satisfy the condition under consideration (with the same $\varepsilon$ ) and with respect to each of them $V$ is the direct sum of two isotropic subspaces, then $f_{1}$ and $f_{2}$ are equivalent.

Problem 37.30

(a)(b)

Let $f$ be a bilinear (sesquilinear) function on a space $V$ , and suppose that for any $x, y \in V$ , $f(x, y)=0$ implies $f(y, x)=0$ . Prove that:

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(a)

if $f$ is a bilinear function, then $f$ is either symmetric or skew-symmetric;

(b)

if $f$ is a sesquilinear function, then $f$ is either Hermitian or skew-Hermitian.

Problem 37.31

(a)(b)

Let $f_{1}, f_{2}$ be bilinear (sesquilinear) functions on a space $V$ with basis $\left(e_{1}, \ldots , e_{n}\right)$ . On the space $W$ with basis $\left(a_{11}, a_{12}, \ldots , a_{1 n}, a_{21}, \ldots , a_{2 n}, \ldots , a_{n 1}, \ldots , a_{n n}\right)$ , define a bilinear function $f$ by setting

f\left(a_{i j}, a_{k l}\right)=f_{1}\left(e_{i}, e_{k}\right) f_{2}\left(e_{j}, e_{l}\right)

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(a)

Find the matrix of the function $f$ in the given basis.

(b)

Prove that if the space $V$ is the direct sum of totally isotropic subspaces with respect to $f_{1}$ , then $W$ is the direct sum of totally isotropic subspaces with respect to $f$ .

Problem 37.32

(a)(b)

Without performing computations, determine whether the following bilinear functions are equivalent:

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(a)

$f_{1}(x, y)=2 x_{1} y_{2}-3 x_{1} y_{3}+x_{2} y_{3}-2 x_{2} y_{1}-x_{3} y_{2}+3 x_{3} y_{1}$ , $f_{2}(x, y)=x_{1} y_{2}-x_{2} y_{1}+2 x_{2} y_{2}+3 x_{1} y_{3}-3 x_{3} y_{1}$ ;

(b)

$f_{1}(x, y)=x_{1} y_{1}+i x_{1} y_{2}$ , $f_{2}(x, y)=2 x_{1} y_{1}+(1+i) x_{1} y_{2}+(1-i) x_{2} y_{1}-i x_{2} y_{2}$ .

Problem 37.33

(a)(b)(c)(d)

Reduce the following skew-symmetric bilinear functions to canonical form:

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(a)

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

(b)

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

(c)

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

(d)

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

Problem 37.34

(a)(b)

Reduce the skew-Hermitian function on a complex space to canonical form:

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(a)

$x_{1} \bar{y}_{2}-i x_{1} \bar{y}_{3}-i x_{2} \bar{y}_{3}-\bar{x}_{1} y_{2}+i \bar{x}_{1} y_{3}+i \bar{x}_{2} y_{3}+i x_{1} \bar{y}_{1}-i \bar{x}_{1} y_{1}$ ;

(b)

$(1+i) x_{1} \bar{y}_{2}+2 x_{1} \bar{y}_{3}+i x_{1} \bar{y}_{4}-(1-i) x_{2} \bar{y}_{3}-(1-i) \bar{x}_{1} y_{3}+i \bar{x}_{1} y_{4}+(1+i) \bar{x}_{2} y_{3}+2 i x_{2} \bar{y}_{2}-2 \bar{x}_{1} y_{3}$ .

Problem 37.35

Prove that the function $h(f, g)=\int_{0}^{1} f g^{\prime } d x$ on the space of polynomials of degree $\leqslant 5$ vanishing at the points 0 and 1 is skew-symmetric, and find a canonical basis for it.

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Problem 37.36

Prove that the determinant of an integer skew-symmetric matrix is the square of an integer.

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Problem 37.37

Let $f$ be a skew-symmetric bilinear function on a space $V$ , let $W$ be a subspace of $V$ , and let $W^{\perp }$ be its orthogonal complement with respect to $f$ . Prove that $\operatorname {dim} W-\operatorname {dim}\left(W \cap W^{\perp }\right)$ is an even number.

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Problem 37.38

Prove that for any skew-Hermitian complex matrix $A$ there is a nonsingular matrix $C$ such that $C A^{t} \bar{C}$ is a diagonal matrix, with purely imaginary complex numbers on the main diagonal.

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Problem 37.39

Let $f$ be a skew-symmetric bilinear function on a space $V$ , let $V^{\prime }$ be its kernel, and let $W$ be a maximal totally isotropic subspace. Prove the equality

\operatorname {dim} W=\frac{\operatorname {dim} V+\operatorname {dim} V^{\prime }}{2}

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Problem 37.40

Let $f$ be a nondegenerate skew-symmetric bilinear function on an $n$ -dimensional space $V$ , and let $G=\left(g_{i j}\right)$ be a skew-symmetric matrix of order $n$ . Prove that there exist vectors $v_{1}, \ldots , v_{n} \in V$ such that $g_{i j}=f\left(v_{i}, v_{j}\right)$ .

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Problem 37.41

Let $f(x, y)$ be a Hermitian function on a complex space, $q(x)=f(x, x)$ . Prove the equality

4 f(x, y)=q(x+y)-q(x-y)+i q(x+i y)-i q(x-i y)

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Problem 37.42

Prove that the real and imaginary parts of a Hermitian function on a complex vector space $V$ are, respectively, a symmetric and a skew-symmetric function on $V$ regarded as a $2 n$ -dimensional real vector space.

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Problem 37.43

Prove that if $f$ is a positive-definite Hermitian form on a complex space, then

f(x, y) \overline{f(x, y)} \leqslant f(x, x) \cdot f(y, y)

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Problem 37.44

Let $\mathcal{A}$ be a linear operator and $f$ a positive definite Hermitian function on a complex vector space $V$ . Prove that if $f(\mathcal{A}(x), x)=0$ for every $x \in V$ , then $\mathcal{A}$ is the zero operator.

Is this statement true for symmetric bilinear functions on a real space $V$ ?

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Problem 37.45

(a)(b)

For which values of $n$ can a nondegenerate bilinear function on an $n$ -dimensional vector space have:

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(a)

a totally isotropic subspace of dimension $n-1$ ;

(b)

a totally isotropic subspace of dimension $n-2$ ?

Derive a formula for the maximum possible dimension of a totally isotropic subspace.

Problem 37.46

(a)(b)

Let $A=\left(a_{i j}\right) \in \mathbf{M}_{n}(\mathbb {R})$ be a symmetric matrix and let

L(f)=\sum _{i, j=1}^{n} a_{i j} \frac{\partial ^{2} f}{\partial x_{i} \partial x_{j}}

be a differential operator on $\mathbb {R}\left[x_{1}, \ldots , x_{n}\right]$ .

Prove that:

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(a)

C\left[\begin{smallmatrix} x_{1} \\ \vdots \\ x_{n} \end{smallmatrix}\right]=\left[\begin{smallmatrix} y_{1} \\ \vdots \\ y_{n} \end{smallmatrix}\right], \quad C \in \mathbf{G L}_{n}(\mathbb {R})

is a change of variables, then

L(f)=\sum _{i, j=1}^{n} b_{i j} \frac{\partial ^{2} f}{\partial y_{i} \partial y_{j}},

where $\left(b_{i j}\right)=C A^{t} C$ ;

(b)

there exists a nondegenerate linear change of variables in $\mathbb {R}\left[x_{1}, \ldots , x_{n}\right]$ such that, in terms of the new variables $y_{1}, \ldots , y_{k}$ ,

L(f)=\frac{\partial ^{2} f}{\partial y_{1}^{2}}+\ldots +\frac{\partial ^{2} f}{\partial y_{k}^{2}}-\frac{\partial ^{2} f}{\partial y_{k+1}^{2}}-\ldots -\frac{\partial ^{2} f}{\partial y_{s}^{2}}

where $0 \leqslant k \leqslant s \leqslant n$ .

Section 38Symmetric bilinear, Hermitian and quadratic functions[35/63%]

Problem 38.1

Which of the bilinear functions from problem 37.1 are symmetric?

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Problem 38.2

(a)(b)(c)

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(a)

Which of the sesquilinear functions from problem 37.5 are Hermitian?

(b)

Is the function $f(x, y)$ from 37.18(b) Hermitian?

(c)

Is the function $f(u, v)$ from 37.20(b) Hermitian?

Problem 38.3

Without performing any calculations, determine whether the bilinear functions $f_{1}(x, y)=x_{1} y_{1}+2 x_{1} y_{2}+3 x_{1} y_{3}+4 x_{2} y_{1}+5 x_{2} y_{2}+6 x_{2} y_{3}+7 x_{3} y_{1}+8 x_{3} y_{2}+10 x_{3} y_{3}$ , $f_{2}(x, y)=2 x_{1} y_{1}-x_{1} y_{3}+x_{2} y_{2}-x_{3} y_{1}+5 x_{3} y_{3}$ .

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Problem 38.4

(a)(b)

Without performing any calculations, determine for which of the bilinear functions $f$ there exists a basis in which the matrix of this function is diagonal:

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(a)

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

(b)

$-x_{1} y_{2}-x_{2} y_{1}+3 x_{2} y_{2}+5 x_{2} y_{3}+5 x_{3} y_{2}-x_{3} y_{3}$ .

Problem 38.5

(a)(b)(c)

Prove that for orthogonal complements of subspaces with respect to a nondegenerate symmetric (Hermitian) function, the following equalities hold:

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(a)

$\left(U^{\perp }\right)^{\perp }=U$ ;

(b)

$\left(U_{1}+U_{2}\right)^{\perp }=U_{1}^{\perp } \cap U_{2}^{\perp }$ ;

(c)

$\left(U_{1} \cap U_{2}\right)^{\perp }=U_{1}^{\perp }+U_{2}^{\perp }$ .

Problem 38.6

(a)(b)

Find the orthogonal complement of the linear span $\left\langle f_{1}, f_{2}\right\rangle$ with respect to the bilinear function with matrix $F$ , if:

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(a)

$F=\left(\begin{array}{rrr}1 & -1 & 2 \\ -1 & 0 & -3 \\ 2 & -3 & 7\end{array}\right), \quad f_{1}=(1,2,3), \quad f_{2}=(4,5,6) ;$

(b)

$F=\left(\begin{array}{rrr}-1 & 2 & 5 \\ 2 & 2 & 8 \\ 5 & 8 & 29\end{array}\right), \quad f_{1}=(-3,-15,21) ; \quad f_{2}=(2,10,-14)$ .

Problem 38.7

(a)(b)

Find the orthogonal complement of the linear span $\left\langle e_{1}, e_{2}\right\rangle$ with respect to the Hermitian function with matrix $G$ , if:

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(a)

$G=\left(\begin{array}{ccc}1 & i & 1-i \\ -i & 0 & -2 \\ 1+i & -2 & -2\end{array}\right), \quad \begin{aligned} & e_{1}=(i, 1,-1), \\ & e_{2}=(1-2 i,-i, 3) ;\end{aligned}$

(b)

$G=\left(\begin{array}{ccc}0 & -2+i & -i \\ -2-i & 2 & -1+i \\ i & -1-i & -1\end{array}\right), \quad \begin{array}{ll}e_{1} & =(-i+1,2,0), \\ e_{2} & =(-1+3 i,-3 i, 2).\end{array}$

Problem 38.8

(a)(b)

Using Jacobi's method, find the canonical form of the symmetric bilinear functions:

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(a)

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

(b)

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

Problem 38.9

(a)(b)

Using Jacobi's method, determine the equivalence of the bilinear functions with matrices

\left[\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 0 & -1 \\ 3 & -1 & 3 \end{smallmatrix}\right], \quad \left[\begin{smallmatrix} 1 & 3 & 0 \\ 3 & 1 & 1 \\ 0 & 1 & 5 \end{smallmatrix}\right]:

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(a)

over the field of real numbers;

(b)

over the field of rational numbers.

Problem 38.10

Which of the symmetric bilinear functions from problems 37.1 and 37.5 are positive definite?

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Problem 38.11

(a)(b)(c)(d)(e)(f)

For which values of $\lambda$ are the following quadratic functions positive definite:

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(a)

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

(b)

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

(c)

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

(d)

$x_{1}^{2}+4 x_{2}^{2}+x_{3}^{2}+2 \lambda x_{1} x_{2}+10 x_{1} x_{3}+6 x_{2} x_{3}$ ;

(e)

$x_{1} \bar{x}_{1}+i x_{1} \bar{x}_{2}-i x_{2} \bar{x}_{1}+\lambda x_{2} \bar{x}_{2}$ ;

(f)

$2 x_{1} \bar{x}_{1}-(1-i) x_{1} \bar{x}_{3}-(1+i) \bar{x}_{1} x_{3}+2 \lambda x_{2} \bar{x}_{3}+2 \bar{\lambda } \bar{x}_{2} x_{3}+x_{2} \bar{x}_{2}+5 x_{3} \bar{x}_{3}$ ?

Problem 38.12

Prove that for any positive definite symmetric bilinear (Hermitian) function $f$ the inequality

\sqrt{f(x+y, x+y)} \leqslant \sqrt{f(x, x)}+\sqrt{f(y, y)}

holds, with equality if and only if $\alpha x=\beta y$ , where $\alpha , \beta$ are nonnegative real numbers not both zero.

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Problem 38.13

Without using Sylvester's criterion, prove that for the positive definiteness of the quadratic function $\sum_{i, j=1}^{n} a_{i j} x_{i} \bar{x}_{j}$ the condition $a_{i i}>0(i=1, \ldots , n)$ is necessary but not sufficient.

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Problem 38.14

(a)(b)(c)(d)

For which values of $\lambda$ are the quadratic functions negative definite:

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(a)

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

(b)

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

(c)

$\lambda x_{1} \bar{x}_{1}+3 x_{2} \bar{x}_{2}-i x_{1} \bar{x}_{2}+i \bar{x}_{1} x_{2}$ ;

(d)

$4 x_{1} \bar{x}_{1}-2 x_{2} \bar{x}_{2}-(\lambda +i) x_{1} \bar{x}_{2}-(\bar{\lambda }-i) \bar{x}_{1} x_{2}$ ?

Problem 38.15

(a)(b)(c)(d)

Find the symmetric bilinear (Hermitian) function associated with the quadratic function:

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(a)

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

(b)

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

(c)

$x_{1} \bar{x}_{1}+i x_{1} \bar{x}_{2}-i \bar{x}_{1} x_{2}+2 x_{2} \bar{x}_{2}$ ;

(d)

$(5-i) x_{1} \bar{x}_{2}+(6+i) \bar{x}_{1} x_{2}+x_{2} \bar{x}_{2}$ .

Problem 38.16

(a)(b)

Find the symmetric bilinear function associated with the quadratic function $q(x)=f(x, x)$ , where:

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(a)

$f(x, y)=2 x_{1} y_{1}-3 x_{1} y_{2}-4 x_{1} y_{3}+x_{2} y_{1}-5 x_{2} y_{3}+x_{3} y_{3}$ ;

(b)

$f(x, y)=-x_{1} y_{2}+x_{2} y_{1}-2 x_{2} y_{2}+3 x_{2} y_{3}-x_{3} y_{1}+2 x_{3} y_{3}$ .

Problem 38.17

(a)(b)

Are the quadratic functions equivalent over the field of real numbers:

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(a)

$x_{1}^{2}-2 x_{1} x_{2}+2 x_{3}^{2}+4 x_{2} x_{3}+5 x_{3}^{2} \quad$ and $\quad x_{1}^{2}-4 x_{1} x_{2}+2 x_{1} x_{3}+4 x_{2}^{2}+x_{3}^{2}$ ;

(b)

$2 x_{1}^{2}+9 x_{2}^{2}+3 x_{3}^{2}+8 x_{1} x_{2}-4 x_{1} x_{3}-10 x_{2} x_{3} \quad$ and $2 x_{1}^{2}+3 x_{2}^{2}+6 x_{3}^{2}-4 x_{1} x_{2}-4 x_{1} x_{3}+8 x_{2} x_{3}?$

Problem 38.18

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)

Find the normal form of the quadratic functions:

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(a)

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

(b)

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

(c)

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

(d)

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

(e)

$x_{1} \bar{x}_{1}-i x_{1} \bar{x}_{2}+i \bar{x}_{1} x_{2}+2 x_{2} \bar{x}_{2}$ ;

(f)

$(1-i) x_{1} \bar{x}_{2}+(1+i) \bar{x}_{1} x_{2}+(1-2 i) x_{1} \bar{x}_{3}+(1+2 i) \bar{x}_{1} x_{3}+x_{2} \bar{x}_{3}+\bar{x}_{2} x_{3}$ ;

(g)

$\sum_{i, j=1}^{n} a_{i} a_{j} x_{i} x_{j}$ , where not all the numbers $a_{1}, \ldots , a_{n}$ equal $0 ;$

(h)

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

(i)

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

(j)

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

(k)

$\sum_{i=1}^{n}\left(x_{i}-s\right)^{2}, \quad s=\frac{x_{1}+\ldots +x_{n}}{n}$ ;

(l)

$\sum_{1 \leqslant i<j \leqslant n}\left|i-j\right| x_{i} x_{j}$ .

Problem 38.19

(a)(b)

Are the quadratic functions equivalent over the field of complex numbers:

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(a)

$x_{1}^{2}-2 x_{1} x_{2}+2 x_{1} x_{3}-2 x_{1} x_{4}+x_{2}^{2}+2 x_{2} x_{3}-4 x_{2} x_{4}+x_{3}^{2}-2 x_{4}^{2}$ and $x_{1}^{2}+x_{1} x_{2}+x_{3} x_{4}$ ;

(b)

$x_{1}^{2}+4 x_{2}^{2}+x_{3}^{2}+4 x_{1} x_{2}-2 x_{1} x_{3}$ and $x_{1}^{2}+2 x_{2}^{2}-x_{3}^{2}+4 x_{1} x_{2}-2 x_{1} x_{3}-4 x_{2} x_{3}$ ?

Problem 38.20

Let $q$ be a map from a real vector space $V$ to the field $\mathbb {R}$ for which there exist quadratic functions $a, b$ and a bilinear function $c$ such that

q(\lambda x+\mu y)=\lambda ^{2} a(x)+\lambda \mu c(x, y)+\mu ^{2} b(y)

for all $\lambda , \mu \in \mathbb {R}$ and $x, y \in V$ . Prove that $q$ is a quadratic function.

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Problem 38.21

Let $f_{1}, \ldots , f_{r+s}$ be linear functions. Prove that the positive index of inertia of the function

q(x)=\left|f_{1}(x)\right|^{2}+\ldots +\left|f_{r}(x)\right|^{2}-\left|f_{r+1}(x)\right|^{2}-\ldots -\left|f_{r+s}(x)\right|^{2}

does not exceed $r$ , and the negative index does not exceed $s$ .

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Problem 38.22

(a)(b)

Find the positive and negative indices of inertia:

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(a)

of the quadratic function $q(x)=\operatorname {tr} x^{2}$ on the space $\mathbf{M}_{n}(\mathbb {R})$ ;

(b)

of the quadratic function $q(x)=\operatorname {tr}(x \bar{x})$ on the space $\mathbf{M}_{n}(\mathbb {C})$ .

Problem 38.23

Let $f$ be a nondegenerate symmetric bilinear (Hermitian) function on a space $V$ of dimension $\geqslant 3$ . Prove that if $f$ is not identically zero on a two-dimensional subspace $U$ , then there exists a three-dimensional subspace $W \supseteq U$ on which the restriction of $f$ is nondegenerate.

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Problem 38.24

Let $f$ be a nondegenerate symmetric bilinear (Hermitian) function with negative index of inertia equal to 1, and $f(v, v)<0$ for some vector $v$ . Prove that the restriction of $f$ to any subspace containing $v$ is nondegenerate.

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Problem 38.25

Let $f$ be a nondegenerate symmetric bilinear (Hermitian) function on a space of dimension $\geqslant 3$ . Prove that every isotropic vector lies in the intersection of two two-dimensional subspaces, on each of which the restriction of $f$ is nondegenerate.

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Problem 38.26

Prove that the dimension of a maximal isotropic subspace with respect to a nondegenerate symmetric bilinear (Hermitian) function equals the smaller of its positive and negative indices of inertia.

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Problem 38.27

Find the positive and negative indices of inertia of a nondegenerate quadratic function on a $2 n$ -dimensional vector space possessing an $n$ -dimensional totally isotropic subspace.

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Problem 38.28

(a)(b)

Let a nondegenerate quadratic function $q$ on a $2 n$ -dimensional space $V$ vanish on an $n$ -dimensional subspace $U$ . Prove that:

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(a)

there exists an $n$ -dimensional subspace $U^{\prime }$ such that

V=U \oplus U^{\prime }, \quad q\left(U^{\prime }\right)=0

(b)

in some basis the function $q$ has the form

x_{1} x_{2}+x_{3} x_{4}+\ldots +x_{2 n-1} x_{2 n}

Problem 38.29

Prove that if in a symmetric matrix some principal minor of order $r$ is nonzero, while all the bordering principal minors of orders $r+1$ and $r+2$ are zero, then the rank of this matrix equals $r$ .

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Problem 38.30

Prove that a real symmetric (complex Hermitian) matrix $A$ can be represented in the form $A={ }^{t} C \cdot C$ (respectively $A={ }^{t} C \cdot \bar{C}$ ), where $C$ is a square matrix, if and only if all the principal minors of $A$ are nonnegative.

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Problem 38.31

Find the dimension of the space of symmetric bilinear functions in $n$ variables.

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Problem 38.32

To each (undirected) graph $\Gamma$ with vertices $v_{1}, \ldots , v_{n}$ associate the quadratic function

q_{\Gamma }(x)=\sum _{i, j=1}^{n} a_{i j} x_{i} x_{j}

by setting

a_{i j}=\left\{ \begin{aligned} 2, & \text{ if } i=j, \\ -1, & \text{ if } v_{i} \text{ and } v_{j} \text{ are joined by an edge, } \\ 0, & \text{ if } v_{i} \text{ and } v_{j} \text{ are not joined by an edge, } \end{aligned}\right\}

and consider the graphs

(the number of vertices of the graph $\Gamma_{n}$ equals $n$ , and of the graph $\widetilde{\Gamma }_{n}$ equals $n+1$ ). Prove that for the graphs $\Gamma_{n}$ the function $q_{\Gamma }$ is positive definite, while for the graphs $\widetilde{\Gamma }_{n}$ it is positive semidefinite: $q_{\Gamma }(x) \geqslant 0$ for every $x$ .

Origin: AI-translated from another node content of the same node

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Problem 38.33

Let $q$ be a nondegenerate quadratic function on a space $V$ over an arbitrary field $F$ . Prove that if there exists a nonzero vector $x \in V$ for which $q(x)=0$ , then the map $q: V \rightarrow F$ is surjective.

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

Problem 38.34

Let $f(x, y)$ be a Hermitian positive semidefinite function, with $f(z, z)=0$ for some $z \neq 0$ . Prove that $f(z, t)=0$ for all $t$ .

Origin: AI-translated from another node content of the same node

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Problem 38.35

Let $f(x, y)$ be a positive definite Hermitian function, with $f(x, x)=f(y, y)=1$ for some $x, y \neq 0$ . Prove that $\left|f(x, y)\right| \leqslant 1$ .

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source