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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 eigenvalues matrices 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
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- V. Yu. Kiselev, A. S. Pyartli, T. F. Kalugina Higher mathematics. Linear algebra. - Ivanovo State Energy University.
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See what “Characteristic polynomial of a matrix” is in other dictionaries:
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Let A- square real or complex matrix of nth order. Matrix
with the variable A, which accepts any numeric values, called characteristic matrix matrices A. Its determinant
represents a polynomial in a variable of degree A p. This polynomial is called characteristic polynomial matrices A.
The fact that the characteristic polynomial is actually a polynomial in the variable A follows directly from the definition of the determinant. Highest degree, equal to n, among all terms of the determinant A - E has the product
The remaining terms of the determinant do not contain at least two matrix elements A- A E with variable A and therefore have a degree no higher p - 2. Therefore, the degree of the polynomial is equal to p. Note that the product (5.9) determines not only the degree of the characteristic polynomial, but also its two terms with higher powers
The free term of the characteristic polynomial coincides with its value at A = 0 and is equal to |A - A E= |L|, i.e. determinant of the matrix A.
So the characteristic polynomial of the matrix A order n has the form (see, p.83 and, p.55):
Where Pk- the sum of the main minors of the A>th order of the matrix A, in particular, Pi= ac + «22 + - - +ftnn - the sum of the elements of the main diagonal of the matrix A, called the trace of this matrix and denoted by Sp A, p p- determinant |L| matrices A.
Roots of the characteristic polynomial |A - HE called characteristic roots or characteristic numbers matrices A. Multiplicity to g characteristic root A* in a characteristic polynomial is called algebraic multiplicity this root. Plenty of everyone characteristic roots matrices in which each characteristic root is repeated as many times as its multiplicity is called spectrum of matrix A. If all the characteristic roots of the matrix are simple (i.e., have unit multiplicity), then the spectrum of the matrix is called simple.
In accordance with Vieta's formulas, the coefficients of the characteristic polynomial are related to the characteristic roots as follows:
From these formulas, in particular, the often used relations follow
According to the last equality, the characteristic polynomial of a matrix has zero characteristic roots if and only if the determinant of this matrix is equal to zero, i.e. when the matrix is singular.
Example 5.5. Calculate the characteristic polynomial of a matrix
Solution. In accordance with the definition of the characteristic polynomial, we obtain:
If we use formula (5.10), we first find
and then write
For methods for calculating the characteristic polynomial, see the appendix at the end of the book.
Theorem 5.7.The characteristic polynomials of such matrices coincide.
> If matrices A And IN similar, then for some non-singular matrix Q equality holds IN = Q~ l AQ. Hence,
To an arbitrary polynomial
instead of the variable L you can substitute a square matrix A order p. As a result, we get the matrix P(A) = in A p + a A p ~ 1 --
N----+ a n _ 1 A + a p E, which is called the value of the polynomial P( L)
at L = A. If for a given matrix A equality is true P(A)= O (the value of the polynomial P( A) with L = A is the zero matrix), then A called matrix, root of the polynomial P( A), and the polynomial P(A) itself is a polynomial annihilated by matrix A.
Theorem 5.8. Every square matrix is the root of some nonzero polynomial.
> The set of all square matrices of order n with elements from the field R there is a linear space over R dimensions n 2. In this linear space any system in which at least n 2+1 elements is linearly dependent. Therefore, the system A p , A p -1 , ..., A, E from p 2 + 1 matrices are linearly dependent, i.e. there is such a set of numbers ao, from, ..., a p 2 , which simultaneously do not vanish, so that the equality
This equality means that the matrix A is the root of the polynomial
The theorem proved actually follows from the following statement.
Theorem 5.9 (Hamilton's theorem - Kaley).
Any square matrix is the root of its characteristic polynomial.
Before proving this theorem, let us introduce the concept X-matrices- a matrix whose elements are polynomials in variable A. Any A-matrix can be represented as a polynomial in variable A, the coefficients of which are square matrices of the appropriate order. For example,
> Let A- square matrix of nth order. Consider the adjoint matrix WITH to the matrix A - E. Its elements are algebraic additions elements of the determinant | A - E|, which are polynomials of degree no higher than p- 1. As noted above, the matrix WITH can be represented in the form
where Ci, C2, ..., C p - some numeric matrices. By the main property of the adjoint matrix (see Section 3.S, Corollary 3.2) we have:
In this equality, we replace the matrix C with the sum (5.11), and the characteristic polynomial with the sum (5.10). Then we get the equality
Opening the parentheses on both sides of the equality and equating the coefficients for equal degrees L, we obtain a system from n+ 1 equalities: 
Let us multiply the first equality of the system by A p, the second - on L p_1, etc., n-e equality - on A, (p+ 1)th equality - on A° = E:
When adding these equalities on the left side we get a zero matrix, and on the right side we get the expression
That's why f(A) = 0. ?
5.6. Characteristic and minimal polynomial
A polynomial 92(A) of minimal degree, having a leading coefficient equal to one and annihilated by the matrix A, called minimal polynomial this matrix.
Theorem 5 . 10 . Any polynomial that is canceled by the matrix A is completely divisible by the minimal polynomial of this matrix. In particular, the characteristic polynomial of a matrix is divided by its minimal polynomial.
O Divide the polynomial P( A) to the minimal polynomial 9?(A) with remainder: P( A) = 99(A) g(A) + g(A), where the polynomial g(A) has a degree less than the degree 92(A). Replacing variable A with a matrix A, we get:
Because P(A)= p(A) = 0 , then G (A) = 0 . But this equality is possible only if the polynomial g (A) null. Otherwise, a contradiction arises with the definition of a minimal polynomial. Equality G = 0 means that the polynomial P( A) is completely divisible by 92(A). ?
Consequence 5 .1 . Any root of the minimal polynomial of a matrix is the root of its characteristic polynomial.
O As established in the proof of the theorem, the characteristic polynomial /(A) is related to the minimal polynomial 92(A) by the equality /(A) = 99(A) q(). The statement of the corollary follows from this equality. ?
Let's note a few more useful facts(cm. [ 7 ], With. 100 ).
Characteristic polynomial | A - HE the matrix A and its minimal polynomial 92(A) are related by the relation
Where Dn- 1 - largest common divisor all minors of the matrix A - A E, having (n - 1 )th order.
The roots of the minimal polynomial 92(A) are all the different roots of the characteristic polynomial | A- A E and if
where 1^ p to ^ t k: k = 1,2
Formula (5.12) allows you to find the minimum polynomial of the matrix. Another way to construct a minimal matrix polynomial is discussed below (see Section 6.5).
Example 5.6. Find the minimum polynomial of a matrix
Solution. In the previous examples for the matrix A characteristic polynomial found A - E= - A 3 + 2 L 2 + L - 2. General greatest divisor D2 all second order minors of the matrix
is equal to one, since its minors
mutually simple. That's why
Example 5.7. Find characteristic and minimal polynomials of matrices
Solution: For a matrix A direct calculation of the determinant we find the characteristic polynomial
Let us write down all the minors of the second order of the matrix A - A E:
Common greatest divisor D2 of all these minors there are A - 4. Therefore, the minimal polynomial of the matrix A has the form:
Note that D2 can be found differently. Indeed, if in the matrix A - E substitute A = 4, we get the matrix
rank G - 1. Consequently, all second-order minors of this matrix are equal to zero. This means that all second order minors of the matrix A - L E are divisible by A - 4, and all these minors cannot be divisible by greater degree binomial A - 4, since, for example, minor
is divisible only by the first power of this binomial. Consequently, ?>2 includes the factor A -4 to the first power. Other multipliers from | A - A?^1 are not included in?>2, since, for example, the second-order minor just written out is not divisible by them. Therefore Dg = A - 4.
For matrix A2 Also, by directly calculating the determinant, we find the characteristic polynomial
second order minors
mutually simple. That's why D2 = 1 and
The considered example shows that different matrices may have the same characteristic but different minimal polynomials.
Considering that the matrices of a given linear operator in different bases are similar and have the same characteristic polynomial, it is logical to call this polynomial characteristic polynomial of a linear operator, and its roots are characteristic roots of a linear operator.
Note also that the transposed matrix A T has the same as the matrix A characteristic polynomials and characteristic numbers.
Eigenvectors and eigenvalues of the linear operator
Let A be a linear operator from . The number is called operator eigenvalue A, if there is a non-zero vector such that A . In this case, the vector is called operator eigenvector A, corresponding to its own value. The set of all eigenvalues of a linear operator A is called its spectrum.
Determinant of the linear operator And detA is called det A, where A is the matrix of the linear operator A in any basis. Polynomial relative l called characteristic polynomial of the operator A. It does not depend on the choice of basis.
Equation
called characteristic(or centuries-old) operator equation A.
In order for the number l was an eigenvalue of the operator A, it is necessary and sufficient that this number be the root of the characteristic equation (7.7) of the operator A.
For identical operator All non-zero vectors of space are eigenvectors (with eigenvalue, equal to one). For zero operator All non-zero vectors of space are eigenvectors (with eigenvalue, equal to zero). The simplest form is taken by the matrix of a linear operator having n linearly independent vectors.
Theorem 7.2. In order for the matrixAlinear operator A was diagonal in the basis, it is necessary and sufficient that the basis vectors be eigenvectors of this operator.
However, not every linear operator in n-dimensional vector space has n linearly independent eigenvectors. A basis of eigenvectors is usually called an “eigenbasis”. Let the eigenvalues 
linear operator A are different. Then the corresponding eigenvectors are linearly independent. Consequently, an “own basis” exists in this case.
So, if the characteristic polynomial of a linear operator A has n different roots, then in some basis the matrix A operator A has a diagonal form.
When finding the eigenvectors of a linear transformation, it should be borne in mind that they are determined up to an arbitrary factor, i.e. if some vector is an eigenvector, then the vector is also an eigenvector. Thus, the proper direction or proper straight line is actually determined, which remains unchanged under a given linear transformation.
Characteristic polynomial
is defined for an arbitrary square matrix as 1) 
, where is an identity matrix of the same order.
Example. For :
Theorem.
Figuratively speaking, the coefficient at is obtained by summing all the minors of the th order of the matrix, built on the elements of its main diagonal.
