Eigenvalues And Eigenvector

Eigenvalues of Matrices:

The eigenvalue are also called characteristics roots, proper values, latent roots, etc. These have fundamental importance is understanding the properties of a matrix. For arbitrary square matrices, eigenvalues are complex root of the characteristics polynomials. After all not all polynomials have real roots. For example

λ2 1 = 0 ahs roots λ1 = i and λ2 = -i where i denotes The eigenvalues are denoted by λi where i = 1,2......,n are in non - increasing order of absolute values. For complex number the absolute value is defined as the square root of the product of the number and its complex conjugate. The complex conjugate of i is -i, their product is 1 with square root also 1. Thus lil = 1 and l -il = 1 and the polynomials holds true.

By the fundamental theorem of algebra an nth degree polynomials defined over the field of complex number has n roots. Hence we have to count each eigenvalue with its proper multiplicity when we write If eigenvalues are all real we distinguish between positive and negative eigenvalues when ordering them by;

Eigenvector of Matrices:

Eigenvector z of an nxn matrix A is of dimension nx1. These are defined by the relationship: Az = λz which can be written as Az = λIz. Now moving Iz to the left hand side we have In the above example where n = 2 the relation is a system of two equations when the matrices are explicity written out. For the 2x2 matrix above the two equation are:

Note that a so-called trival solution is to choose z = 0, that is both elements of z are zero. We ignore this valid solution because it is not interested or useful. This system of n equations is called degeneration and it has no solution unless additional restriction are imposed. The vector z has two element z1 and z2 to be solved from two equations.

Example:- Impose the additional restriction that the z vector must lie on the unit circle that is Now show that the two ways of solving the system of two equations. The choice yields to two solutions. Associated with each of the two solutions there are two eigenvector which satiesfies the defining equations

The characteristics equation (1)  for the above example (n = 2) is,

A remarkable result known as Cayle-Hamilton theorem satates that the matrix A satiesfies its own characteristics equation in the sense that if we replace λ by A, then we have