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 K N is called the characteristic polynomial resulting from the expression:
- P (x) = | A-xI N|
where A is the square matrix associated with T , I 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 K N , for it the following properties are verified:
- The eigenvalues of Tare roots of P (x) = 0 .
- If all the Nroots of P (x) belong to K then they are eigenvalues of T and it is diagonalizable.
- 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 M 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 M 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.