Characteristic polynomial

Characteristic polynomial . It is said of the polynomial of degree N resulting from the equation | A-xI N | where A is the square matrix associated with the linear application T of the same order.

The real roots of the polynomial it will eigenvalues of T .

Hence, the characteristic polynomials are essential for obtaining the values, subspaces and eigenvectors of a linear application. They also verify the existence of said linear values ​​and the diagonalizability of the matrix.

Summary

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  • 1 Definition
  • 2 Properties
  • 3 Examples
  • 4 Importance
  • 5 See also
  • 6 Sources

Definition

Let T be an endomorphic linear transformation or application of order N over N is called the characteristic polynomial resulting from the expression:

  • P (x) = | A-xI N|

where A is the square matrix associated with T , N is the identity matrix of order N .

While it is known as a characteristic equation to the expression of matrix algebra (A-xI N ) = 0 .

Properties

Let be a characteristic polynomial P (x) of the linear application T on N , for it the following properties are verified:

  1. The eigenvalues ​​of Tare roots of P (x) = 0 .
  2. If all the Nroots of P (x) belong to K then they are eigenvalues ​​of T and it is diagonalizable.
  3. Two similar matrices have the same characteristic polynomial.

Examples

Let matrix A be over the set of reals:

associated with the linear application T: R 3 -> R 3 ; obtain the values of A .

1st . We propose | A-kI | = 0 to obtain the characteristic polynomial:

which is reduced to:

  • -k 3+ 6k 2 + 2k-12 = 0 (characteristic polynomial of A )

2nd . The roots of the polynomial are determined:

  • -k 3+ 2k + 6k 2 -12 = 0
  • = -k (k 2-2) +6 (k 2 -2)
  • (6-k) (k 2-2) = 0

being 6 and that are proper values ​​of A because they are all real .

In the case of the matrix 2 (R) :

To determine if it has eigenvalues, the characteristic polynomial is first sought by:

Remaining:

  • P (x) = x 2+ 1 = 0

That as is known has no real solution.

On the other hand, if M were defined on complex numbers ; the solutions i and -i will be eigenvalues ​​of 2 (R) .

Importance

Values ​​and eigenvectors are key to the diagonalization of square matrices, a process that is done by solving the characteristic polynomial of the square matrix associated with the linear transformation in question, generally using the Cayley-Hamilton theorem . For matrices higher than order 3, polynomials will be obtained that will not have a general factorization method.

In any case, the characteristic polynomial makes it possible to link the search for the proper values ​​of a linear application with the resolution of a polynomial of a variable of degree equal to the order of the application, with the drawbacks and advantages of this part of algebra.

 

by Abdullah Sam
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