Math Problems Vault

Problem 1257

(1)(2)(3)

For the quadratic functions $f$ and $g$ , determine whether there exists a linear transformation taking the function $f$ to the function $g$ :

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(1)

$f = 2 x_{1}^{2}+9 x_{2}^{2}+3 x_{3}^{2}+8 x_{1} x_{2}-4 x_{1} x_{3}-10 x_{2} x_{3}$ ,

g = 5 y_{1}^{2}+6 y_{2}^{2}+12 y_{1} y_{2}

(2)

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

g = 4 y_{1}^{2}+y_{2}^{2}+9 y_{3}^{2}-12 y_{1} y_{3}

(3)

$f = x_{1}^{2}+5 x_{2}^{2}+11 x_{3}^{2}+6 x_{4}^{2}+4 x_{1} x_{2}-2 x_{1} x_{3}+2 x_{2} x_{3}-14 x_{3} x_{4}$ , $g = -y_{1}^{2}-3 y_{2}^{2}+9 y_{3}^{2}+6 y_{4}^{4}-4 y_{1} y_{2}+2 y_{1} y_{3}+10 y_{2} y_{3}-4 y_{2} y_{4}- -14 y_{3} y_{4}$ .

Problem 1258

Using Lagrange's method, reduce the quadratic function to normal form and find the matrix of the corresponding linear transformation:

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(1)

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

(2)

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

(3)

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

(4)

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

(5)

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

(6)

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

(7)

$x_{1}^{2}+5 x_{2}^{2}+9 x_{3}^{2}+5 x_{4}^{2}-4 x_{1} x_{2}+2 x_{1} x_{3}+2 x_{1} x_{4}+2 x_{2} x_{3}-8 x_{3} x_{4}$ .

Problem 1259

Show that the symmetric bilinear function on a two-dimensional linear space over a field of characteristic 2, given by the matrix $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$ , cannot be reduced to diagonal form.

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Problem 1260

Let $\varphi (x, y)$ be a nonzero skew-symmetric bilinear function on an $n$ -dimensional linear space. Prove that there exists a basis in which $\varphi (x, y)$ has the canonical form:

\begin{aligned} \varphi (x, y) = x_{1} y_{2}-x_{2} y_{1}+x_{3} y_{4}-x_{4} y_{3}+\ldots \\ \ldots +x_{2 k-1} y_{2 k}-x_{2 k} y_{2 k-1}, \quad 1 \leqslant k \leqslant n / 2 \end{aligned}

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Problem 1261

Reduce the skew-symmetric bilinear function to canonical form (finding the transformation):

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(1)

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

(2)

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

+x_{3} y_{4}-x_{4} y_{3}

(3)

$x_{1} y_{2}-x_{2} y_{1}+3 x_{1} y_{3}-3 x_{3} y_{1}-2 x_{1} y_{4}+2 x_{4} y_{1}-7 x_{2} y_{3}+7 x_{3} y_{2}+$

+3 x_{2} y_{4}-3 x_{4} y_{2}-4 x_{3} y_{4}+4 x_{4} y_{3}

(4)

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

+2 x_{4} y_{2}-x_{4} y_{3}

(5)

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

-2 x_{4} y_{2}+x_{4} y_{3}

Problem 1262

Reduce the following quadratic functions to diagonal form and find the expression of the new coordinates in terms of the old ones:

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(1)

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

(2)

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

(3)

$\sum_{i < j}^{n} x_{i} x_{j}$ ;

(4)

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

(5)

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

Problem 1263

Let a quadratic function of the form

\varphi = l_{1}^{2}+l_{2}^{2}+\ldots +l_{p}^{2}-l_{p+1}^{2}-l_{p+2}^{2}-\ldots -l_{p+q}^{2}

be given, where $l_{i}, i = 1, \ldots , p+q$ , are real linear functions of $x_{1}, \ldots , x_{n}$ . Prove that the positive inertia index of the function $\varphi$ does not exceed $p$ , and the negative inertia index does not exceed $q$ .

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Problem 1264

In the space $\mathbb {R}_{1}[x]$ of polynomials of degree at most 1, a quadratic function is given

\varphi (P) = \int _{0}^{a} P^{2}(x)(x-1) d x

where $a > 0$ . Find the positive inertia index $p$ and negative inertia index $q$ of this quadratic function as functions of $a$ .

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Problem 1265

(1)(2)

Find all values of the parameter $\alpha$ for which the given quadratic form is positive-definite:

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(1)

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

(2)

$2 x_{1}^{2}+x_{2}^{2}+3 x_{3}^{2}+2 \alpha x_{1} x_{2}+2 x_{1} x_{3}$ .

Problem 1266

(1)(2)

Reduce the Hermitian sesquilinear function to normal form (finding the transformation):

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(1)

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

(2)

$\bar{x}_{1} x_{1}+9 \bar{x}_{2} x_{2}+9 \bar{x}_{3} x_{3}+3 i \bar{x}_{1} x_{2}-3 i \bar{x}_{2} x_{1}-3 \bar{x}_{1} x_{3}-3 \bar{x}_{3} x_{1}+$

+7 i \bar{x}_{2} x_{3}-7 i \bar{x}_{3} x_{2}

Problem 1267

(1)(2)

Reduce the skew-Hermitian sesquilinear function to normal form (finding the transformation):

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(1)

$\bar{x}_{1} x_{2}-\bar{x}_{2} x_{1}$

(2)

$i \bar{x}_{1} x_{1}+9 i \bar{x}_{2} x_{2}+9 i \bar{x}_{3} x_{3}-3 \bar{x}_{1} x_{2}+3 \bar{x}_{2} x_{1}-3 i \bar{x}_{1} x_{3}-3 i \bar{x}_{3} x_{1}-$

-7 \bar{x}_{2} x_{3}+7 \bar{x}_{3} x_{2}