How to find the characteristic polynomial of a matrix. Characteristic polynomial and characteristic numbers of the matrix

Let us be given a square matrix of order n. Characteristic matrix matrix A is called a matrix

=with the variable λ taking any numerical values.

The determinant of the matrix is ​​a polynomial n th power of λ. This polynomial is called the characteristic polynomial of the matrix A, equation =0 is its characteristic equation, and its roots https://pandia.ru/text/78/250/images/image008_68.gif" width="15" height="17 src="> are called every non-zero vector X, satisfying the condition https://pandia.ru/text/78/250/images/image010_64.gif" width="19" height="24 src="> – number.

The number is called the eigenvalue of the transformation https://pandia.ru/text/78/250/images/image011_63.gif" width="201" height="75"> (*)

If the eigenvalue is known λ , then all eigenvectors of the matrix A, belonging to this eigenvalue, are found as non-zero solutions of this system. On the other hand, this homogeneous system with a square matrix A–λE has non-zero solutions X if and only if the determinant of the matrix of this system is equal to zero and λ belongs to the field in question R. But this means that λ is the root of the characteristic polynomial and belongs to the field R. Thus, the characteristic numbers of the matrix belonging to the main field, and only they, are its eigenvalues. To find all eigenvalues ​​of a matrix A you need to find all its characteristic numbers and select from them only those that belong to the main field R, and to find all eigenvectors of the matrix A need to find everything non-zero system solutions (*) at each eigenvalue λ matrices A.

Example 1. Find eigenvalues ​​and eigenvectors of a real matrix .

Solution. Characteristic polynomial of a matrix A has the form:

https://pandia.ru/text/78/250/images/image014_58.gif" width="144" height="75 src=">=(multiply (2)th column per number (-2) and add with (1m column) =https://pandia.ru/text/78/250/images/image016_45.gif" width="172" height="75">=(multiply (1)th column per number (-1) and add with (3m column) = =(multiply (1)th line to number (2) and add with (2)th line) = =(multiply (2)th column per number (-2) and add with (3m column) =
.

Thus, the characteristic polynomial has roots λ1=6, λ2=λ3= – 3. All of them are real and therefore are eigenvalues ​​of the matrix A.

At λ=6 system ( A–λE)X=0 looks like https://pandia.ru/text/78/250/images/image021_35.gif" width="57" height="75 src=">..gif" width="153" height="75 src= ">.

Its general solution is X=https://pandia.ru/text/78/250/images/image025_28.gif" width="85" height="27 src=">, it gives a general view of the eigenvectors of the matrix A, belonging to the eigenvalue λ= – 3.

Definition

For a given matrix , , where E - identity matrix, is a polynomial in , which is called characteristic polynomial matrices A(sometimes also the “secular equation”).

The value of the characteristic polynomial is that the eigenvalues ​​of the matrix are its roots. Indeed, if the equation has a non-zero solution, then the matrix is ​​singular and its determinant is equal to zero.

Related definitions

Properties

.

Links

• V. Yu. Kiselev, A. S. Pyartli, T. F. Kalugina Higher mathematics. Linear algebra . - Ivanovo State Energy University.

Wikimedia Foundation. 2010.

• Reference curve
• Harald III (King of Norway)

See what “Characteristic polynomial of a matrix” is in other dictionaries:

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Definition

For a given matrix , , where E- the identity matrix is ​​a polynomial in , which is called characteristic polynomial matrices A(sometimes also the “secular equation”).

The value of the characteristic polynomial is that the eigenvalues ​​of the matrix are its roots. Indeed, if the equation has a non-zero solution, then the matrix is ​​singular and its determinant is equal to zero.

Related definitions

Properties

.

Links

• V. Yu. Kiselev, A. S. Pyartli, T. F. Kalugina Higher mathematics. Linear algebra . - Ivanovo State Energy University.

Wikimedia Foundation. 2010.

See what “Characteristic polynomial of a matrix” is in other dictionaries:

In mathematics, a characteristic polynomial can mean: a characteristic polynomial of a matrix, a characteristic polynomial of a linear recurrent sequence, a characteristic polynomial of an ordinary differential equation.... ... Wikipedia

Matrices over the field K are a polynomial over the field K. The degree of X. m. is equal to the order of the square matrix A, the coefficient b1 is equal to the trace of the matrix. (b1 = tr A = a11+ a 22+ ... +a pp), the coefficient b t is equal to the sum of all main minors of the th order, in particular bn=detA... Mathematical Encyclopedia

This term has other meanings, see Minimum polynomial. A minimal matrix polynomial is a annihilating unitary polynomial of minimal degree. Properties The minimal polynomial divides the characteristic polynomial of the matrix... ... Wikipedia

Main article: Functions of matrices Lambda matrix (λ matrix, matrix of polynomials) is a square matrix whose elements are polynomials over some number field. If there is some matrix element that is a polynomial... Wikipedia

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The eigenvector is shown in red. It, unlike the blue one, did not change direction and length during deformation, therefore it is an eigenvector corresponding to the eigenvalue λ = 1. Any vector parallel to the red vector... ... Wikipedia

Square matrices A and B of the same order are said to be similar if there is a non-singular matrix P of the same order such that: Similar matrices are obtained by specifying the same linear transformation of the matrix in different... ... Wikipedia

A characteristic polynomial is a polynomial that determines the eigenvalues ​​of a matrix. Another meaning: The characteristic polynomial of a linear recurrent is a polynomial. Contents 1 Definition ... Wikipedia

Hamilton Cayley's theorem is a famous theorem from matrix theory, named after William Hamilton and Arthur Cayley. Hamilton Cayley's theorem Any square matrix satisfies its characteristic equation. If... Wikipedia

Consider the square matrix A = ||аik||1n. The characteristic matrix for matrix A is called matrix LE-A.

l - a 11 -a 12 ... -a 1n

lE-A = -a21 l - a22 ... -a 2n

….…………………… .

A n1 -a n2 ... l - ann

Determinant of the characteristic matrix

?(l) = |le-A| = |l dik - aik|1n

is a scalar polynomial with respect to l and is called the characteristic polynomial of the matrix A.

We will call the matrix B(l) = ||bik (l)||1n, where bik (l) is the algebraic complement of the element ldik - аik in the determinant?(l), the adjoint matrix for the matrix A.

To find the leading terms of the characteristic polynomial, we use the fact that the value of the determinant is equal to the sum of the products of its elements, taken one from each row and each column and equipped with the appropriate signs. Therefore, in order to obtain the term that has the highest degree relative to l, it is necessary to take the products of the elements of the highest degree. In our case, such a product will be only one product of diagonal elements (l - a11) (l - a22) ... (l - ann). All other products included in the determinant have a degree no higher than n-2, since if one of the factors of such a product is aik (i ? k), then this product will not contain factors l-aii, l-acc and will, therefore , degrees no more than n-2. Thus, ?(l) = (l - a11) ... (l - am) + terms of degree not higher than n-2, or

?(l) = ln - (a11 + … + ann) ln-1 + …(22)

The sum of the diagonal elements of a matrix is ​​called its trace. Formula (22) shows that the degree of the characteristic polynomial of a matrix is ​​equal to the order of this matrix, the leading coefficient of the characteristic polynomial is 1, and the coefficient of ln-1 is equal to the trace of the matrix taken with the opposite sign.

Theorem 3. The characteristic polynomials of similar matrices are equal to each other.

It follows from this theorem, in particular, that similar matrices have identical traces and determinants, since the trace and determinant of a matrix, taken with the appropriate signs, are the coefficients of its characteristic polynomial.

The roots of the characteristic polynomial of a matrix are called its characteristic numbers or eigenvalues. The multiple roots of the characteristic polynomial are called the multiple eigenvalues ​​of the matrix. It is known that the sum of all real and complex roots of a polynomial of degree n, having a leading coefficient of 1, is equal to the coefficient of the (n-1)th degree of the variable taken with the opposite sign. Formula (22) therefore shows that in the field of complex numbers the sum of all eigenvalues ​​of a matrix is ​​equal to its trace.

Theorem of Hamilton and Caelie. Each matrix is ​​the root of its characteristic polynomial, i.e. ?(A)= 0.

?(l) = l - 2 -1 = lI - 5l + 7,

?(A) = AI - 5A + 7E = 3 5 -5 2 1 +7 1 0 = 0 0 = 0.