EIGENVALUES and EIGENVECTORS
 Definitions

Given an
matrix A, some
and
,
with
 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, B_{k} is a basis for the
eigenvectors for ,
and
,
then B is a linearly independent set in R^{n}, 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
20000706