Question

Find the Eigen values and eigen vectors of the system of equations B [5 0- 15]
[-3-4 9]
[5 0-15]

Answer 1

Step-by-step explanation:

Eigenvalues and Eigenvectors

Let A be an n� n matrix over a field F . We recall that a scalar l Î F is said to be an eigenvalue (characteristic value, or a latent root) of A, if there exists a nonzero vector x such that Ax = l x, and that such an x is called an eigen-vector (characteristic vector, or a latent vector) of A corresponding to the eigenvalue l and that the pair (l , x) is called an eigen-pair of A. If l is an eigenvalue of A, the equation: (l I-A)x = 0, has a non-trivial (non-zero) solution and conversely. Thus, this being a homogeneous equation, it follows that l is an eigenvalue of A iff |l I-A| = 0. The expression

fA(x) = |xI-A|

is a monic (the coefficient of the highest power of x in it is 1) polynomial in x of degree n. It is known as the characteristic polynomial of A. Thus l is an eigen value of A iff l is a zero (or root) of the characteristic polynomial fA(x) in F. The equation

fA(x) = 0,

is called the characteristic equation of A. Note that the coefficients of the characteristic polynomial are elements of field F under consideration. If F is an algebraically closed field (for instance, ℂ) we know that a polynomial of precise degree n has precisely n roots counted after multiplicities. Thus, for instance, a complex n� n matrix has n eigen values counted after multiplicities.

Note that l = 0 is an eigenvalue of A iff |A| = 0, i.e., A is a singular matrix. If an eigenvalue l of A is known, the corresponding eigenvector(s) may be obtained by elementary row operations (ero's) performed on the matrix part of the augmented matrix [A-l I|0].

Example: Computation of eigen-values and eigen-vectors of a 2� 2 complex matrix:

The algebraic multiplicity of an eigenvalue l of A is the highest k such that (x-l )k is a factor of fA(x). The geometric multiplicity of an eigenvalue l of a matrix A is the maximum number of linearly independent eigen vectors x of A associated with the eigenvalue l , which is the same as the dimension of the eigenspace of A associated with the eigenvalue l consisting of all x such that Ax = l x. This eigenspace is the same as N(A-l I).

The geometric multiplicity of the eigenvalue 0 of is 2, while that of the eigenvalue 0 of is 1. The algebraic multiplicities of the eigenvalue 0 of both these matrices equal 2.