EIGENVALUES and EIGENVECTORS
matrix A, some
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
A = PBP-1 for some invertible P, then
A is similar to B .
- Similar matrices have the same eigenvalues.
A = PBP-1 and
- A square matrix A is diagonalizable
if A is
similar to a diagonal matrix D (
A = PDP-1).
- Let A be an
- A is diagonalizable iff A has n linearly independent eigenvectors;
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 ,
then B is a linearly independent set in Rn, and
A is diagonalizable iff
dim(Span(B)) = n.
- Diagonalization Method:
- For an
- find the eigenvalues for A,
- find the eigenvectors for A,
Alan C Genz