21-25 емтихан сурактары

21. Quadratic forms. Reducing a quadratic form to a canonical type.

A QF of real variables x₁,x₁,…,x_n is a polynomial of the 2^nd degree according to these variables which doesn’t contain a free term and terms of the 1^st degree. If n=2, then f(x₁,x₂)=a₁₁x₂+2a₁₂x₁x₂+a₂₂x₂. A matrix at which is called matrix of quadratic form and the corresponding determinant – determinant of this quadratic form. Since A is a symmetric matrix then the roots and of the characteristic equation are real numbers.

Let

be normalized eigenvectors corresponding to characteristic numbers in an orthonormal basis The vectors form an orthonormal basis.

_{The matrix} is the transition matrix from the basis to the basis

Formulas of transformation of coordinates at transition to the new orthonormal basis have the following form:

_{Transforming by these formulas the quadratic form} we obtain the following quadratic form:

_{It does not contain terms with}

We say that a quadratic form has been reduced to the canonical type by an orthogonal transformation B.

22. Definite quadratic forms. Criterion of Sylvester.

A quadratic form is called positive definite (negative definite) if for all values the condition () holds and only for

For example, is positive definite; is negative definite. Positive definite and negative definite quadratic forms are called definite.

A quadratic form is called quasi-definite (either non-negative or non-positive) if it takes either only non-negative values or non-positive values, but it takes 0 not only for

The determinants

are called angular minors of a matrix

1. A quadratic form is positive definite if and only if all the angular minors of its matrix are positive.

2. A quadratic form is negative definite if and only if the signs of angular minors alternate as follows:

23. Reducing an equation of curve of the second order to a canonical type.

Let an equation of a curve of the second order in rectangular system of coordinates be given: Consider the quadratic form connected with the equation (1): .Reduce the quadratic form to canonic type by an orthogonal transformation of variables: are eigen-values of the matrix , and the columns of the matrix are orthogonal normalized eigen-vectors (columns) of the matrix . The matrix by properties of orthogonal matrices has the following type: Using the formulas (2), express the linear terms of the equation (1) by the coordinates . In result in the system the equation of the curve takes the following type: . Further, extracting complete squares on both variables by parallel transfer of axes of coordinates of the system pass to the system in which the equation of the curve has canonic type. An equation of a surface of the second order can be reduced to canonic type by analogy

24. Unitary space. Gram matrix, Hermitian matrix, unitary matrix.

A complex linear space where the following conditions hold:

1) ;

2) A complex number z is real iff ;

3) The number is always real and nonnegative;

4) , .

is called unitary iff every ordered pair of elements x and y is put in correspondence a complex number (x,y) called their scalar product so that the following conditions hold:

(1) ;

(2) for every complex number ;

(3) ;

(4) is a real nonnegative number, and .

Let in a basis be given. The scalar product of elements and is presented as , where are components of the matrix named the basis matrix of Gram.

A matrix satisfying the property is called Hermitian. A matrix satisfying the properties and is called unitary.

25. Linear operators in a unitary space.

A linear operator acting in a unitary space is called Hermitian conjugate to a linear operator if for all the following holds: .

Theorem 1. For linear operators and acting in a unitary space the following holds:

and

Proof: Prove the first assertion. We have for all , and consequently .

Similarly, for all and every complex number . 

Theorem 2. The matrix of an operator that is Hermitian conjugate to an operator in in a basis

is defined by the following equality: .

A linear operator acting in a unitary space is called Hermitian self-conjugate (or just Hermitian) if .

Properties of Hermitian operators:

1. Eigen-values of a Hermitian operator are real numbers.

2. Eigen-vectors corresponding to distinct eigen-values of a Hermitian operator are orthogonal.

3. For every Hermitian operator there exists an orthonormal basis consisting of its eigen-vectors.

4. In an orthonormal basis of a unitary space a Hermitian operator has a Hermitian matrix.

An eigen-value of a linear operator is called degenerate if an invariant eigen-subspace corresponding to it has the dimension greater than 1.

A linear operator acting in a unitary space is called unitary (or isometric) if for all the following holds: .

Properties of unitary operators:

1. .

2. If is a unitary operator then there exists the inverse operator that is also unitary and .

3. A unitary operator transfers an orthonormal basis in an orthonormal basis, and conversely if a linear operator transfers an orthonormal basis to an orthonormal basis then is unitary.

4. If is an eigen-value of a unitary operator then .