Eigenvalues and Eigenvectors

Background

We will now review some ideas from linear algebra. Proofs of the theorems are either left as exercises or can be found in any standard text on linear algebra.We know how to solve n linear equations in n unknowns.It was assumed that the determinant of the matrix was nonzero and hence that the solution was unique. In the case of a homogeneous system AX = 0, the unique solution is the trivial solution X = 0.If,there exist nontrivial solutions to AX = 0. Suppose that,and consider solutions to the homogeneous linear
system

A homogeneous system of equations always has the trivial solution .Gaussian elimination can be used to obtain the reduced row echelon form which will be used to form a set of relationships between the variables, and a non-trivial solution.

Example 1.Find the nontrivial solutions to the homogeneous system

Background for Eigenvalues and Eigenvectors

Definition (Linearly Independent).The vectorsare said to be linearly independent if the equation



implies that .If the vectors are not linearly independent they are said to be linearly dependent.

Two vectors in are linearly independent if and only if they are not parallel.Three vectors in are linearly independent if and only if they do not lie in the same plane.

Definition (Linearly Dependent).The vectorsare said to be linearly dependent if there exists a set of numbers not all zero, such that

.

The vectorsare linearly dependent if and only if at least one of them is a linear combination of the others.

A desirable feature for a vector space is the ability to express each vector as s linear combination of vectors chosen from a small subset of vectors.This motivates the next definition.

Definition (Basis).Suppose that is a set ofmvectors in .The setS i s called a basis forif for every vectorthere exists a unique set
of scalars so thatXcan be expressed as the linear combination

Theorem.In,any set ofnlinearly independent vectors forms a basis of.Each vectoris uniquely expressed as a linear combination of the basis vectors.

Theorem.Letbe vectors in.

(i)Ifm>n,then the vectors are linearly independent.

(ii)Ifm=n,then the vectors are linearly dependent if and only if,where.

Applications of mathematics sometimes encounter the following questions:What are the singularities of,whereis a parameter?What is the behavior of the sequence of vectors?What are the geometric features of a linear transformation?Solutions for problems in many different disciplines, such as economics, engineering, and physics, can involve ideas related to these equations. The theory of eigenvalues and eigenvectors is powerful enough to help solve these otherwise intractable problems.

LetAbe a square matrix of dimensionn � nand letXbe a vector of dimensionn.The productY = AXcan be viewed as a linear transformation fromn-dimensional space into itself.We want to find scalarsfor which there exists a nonzero vectorXsuch that

(1);

that is, the linear transformationT(X) = AXmapsXonto the multiple.When this occurs, we callXan eigenvector that corresponds to the eigenvalue, and together they form the eigenpair forA.In general, the scalarand vectorXcan involve complex numbers.For simplicity, most of our illustrations will involve real calculations.However, the techniques are easily extended to the complex case.The n � n identity matrixIcan be used to write equation (1) in the form

(2).

The significance of equation (2) is that the product of the matrixand the nonzero vectorXis the zero vector!The theorem of homogeneous linear system says that (2) has nontrivial solutions if and only if the matrixis singular, that is,

(3).

This determinant can be written in the form

(4)

Definition (Characteristic Polynomial).When the determinant in (4) is expanded, it becomes a polynomial of degree n, which is called the characteristic polynomial

(5)