EIGENVALUES and EIGENVECTORS

Definitions

Given an matrix A, some and , with • is an eigenvalue of A.
• is an eigenvector of A corresponding to .
• The for some is .

Properties
• is an eigenvalue of A iff the equation has a nontrivial solution.
• Diagonal entries of triangular matrices are eigenvalues.
• A is invertible iff 0 is not an eigenvalue for A.
• Eigenvectors for distinct eigenvalues are independent.

The CHARACTERISTIC EQUATION

Characteristic Polynomial
• is an eigenvalue for A iff .
• The roots of characteristic polynomial , are eigenvalues for A.
• has degree n with n (possibly complex) roots.
• The multiplicity(algebraic) of an eigenvalue is the number of times the associated root of is repeated.

Similarity
• If A = PBP-1 for some invertible P, then
A is similar to B .
• Similar matrices have the same eigenvalues.
• If A = PBP-1 and ,
then with .

DIAGONALIZATION

Definition:
A square matrix A is diagonalizable
if A is similar to a diagonal matrix D ( A = PDP-1).

Theorems:
Let A be an matrix.
• A is diagonalizable iff A has n linearly independent eigenvectors;
if A = PDP-1 for a diagonal matrix D, then the diagonal entries of D are eigenvalues for A with eigenvectors in the respective columns of P.
• If A has n distinct eigenvalues, then A is diagonalizable.
• If A has p distinct eigenvalues, Bk is a basis for the eigenvectors for , and ,
then B is a linearly independent set in Rn, and
A is diagonalizable iff dim(Span(B)) = n.

Diagonalization Method:
For an matrix A
1.
find the eigenvalues for A, ,
2.
find the eigenvectors for A, , and
3.
construct and Alan C Genz
2000-07-06