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Eigenvalues and Eigenvectors

MATH 322, Linear Algebra I

J. Robert Buchanan

Department of Mathematics

Spring 2007

Introduction

We have been briefly acquainted with eigenvalues and eigenvectors for 2 × 2 matrices. We have also examined the geometric properties of eigenvalues and eigenvectors for 2 × 2 and 3 × 3 linear transformations. Today we extend these ideas to general n × n matrices and linear transformations.

Eigensystems

Definition If A is an n × n matrix and if there exists a nonzero vector x ∈ R n^ and a scalar λ such that A x = λ x then λ is an eigenvalue of A with corresponding eigenvector x.

Remark: geometrically an eigenvalue/eigenvector pair means A x is a dilation/reflection of x by scalar λ.

Finding Eigenvalues

We must find nonzero solutions of

A x = λ x A x − λ x = 0 λ x − A x = 0 λ I x − A x = 0 (λ I − A ) x = 0 =⇒ det(λ I − A ) = 0

Remarks: The expression det(λ I − A ) is called the characteristic polynomial of A. The expression det(λ I − A ) = 0 is called the characteristic equation of A.

Characteristic Polynomial and Equation

If A is an n × n matrix then

det(λ I − A ) = λ n^ + c 1 λ n −^1 + · · · + cn = p (λ)

where p is a polynomial.

Observation: An n × n matrix will have n eigenvalues. Some of these may be complex numbers.

Example

Find the eigenvalues of A =

Characteristic Polynomial and Equation

If A is an n × n matrix then

det(λ I − A ) = λ n^ + c 1 λ n −^1 + · · · + cn = p (λ)

where p is a polynomial.

Observation: An n × n matrix will have n eigenvalues. Some of these may be complex numbers.

Example

Find the eigenvalues of A =

Eigenvalues and Triangular Matrices

Theorem If A is an n × n triangular matrix, then the eigenvalues of A are the diagonal entries of A.

Proof. The determinant of a triangular matrix is the product of the diagonal entries.

Example

Example Find the eigenvalues of   

Eigenspaces

Remark: The eigenvectors of A x = λ x form a vector space called the eigenspace.

Example

Previously we found the eigenvalues of A =

 (^) to

be λ 1 = 0 and λ 2 = λ 3 = 1. Find the bases of the eigenspaces corresponding to each eigenvalue.

Eigenspaces

Remark: The eigenvectors of A x = λ x form a vector space called the eigenspace.

Example

Previously we found the eigenvalues of A =

 (^) to

be λ 1 = 0 and λ 2 = λ 3 = 1. Find the bases of the eigenspaces corresponding to each eigenvalue.

Powers of a Matrix

Theorem If k is a positive integer and λ is an eigenvalue of A with corresponding eigenvector x , then λ k^ is an eigenvalue of Ak with x as its corresponding eigenvector.

Proof. By induction on k.

Eigenvalues and Invertibility

Theorem A square matrix A is invertible if and only if 0 is not an eigenvalue of A.

Proof. (^1) Suppose A is invertible... (^2) Suppose 0 is not an eigenvalue of A...

Eigenvalues and Invertibility

Theorem A square matrix A is invertible if and only if 0 is not an eigenvalue of A.

Proof. (^1) Suppose A is invertible... (^2) Suppose 0 is not an eigenvalue of A...

Equivalent Statements

Theorem If A is an n × n matrix and if TA : R n^ → R n^ is multiplication by A, then the following are equivalent. (^1) A is invertible. (^2) A x = 0 has only the trivial solution. (^3) The reduced row echelon form of A is In. (^4) A is expressible as the product of elementary matrices. (^5) A x = b is consistent for every n × 1 matrix b. (^6) A x = b has exactly one solution for every n × 1 matrix b. (^7) det( A ) 6 = 0_._ (^8) The range of TA is R n. (^9) TA is one-to-one.