let be an matrix, if there is a non-zero vector , and a (which is allowed to be ) such that , then is an eigenvalue of , and is an eigenvector of

example:

example:

if is an eigenpair of , then so is for scalar

if is invertible, then is an eigenpair of

if is also an eigenpair of , then so is for any non-vanishing linear combinations

if are both eigenpairs of and , then are linearly independent

if , then ,

is a polynomial of

eigenvalues of a triangular matrix are its main digaonal elements, eigenvalues and coincide, eigenvalues of take the form

find eigenvalues of

find eigenvalues of following matrices

let , then is called the characteristic polynomial of

if is a root of multiplicity , then it corresponds to at most linearly independent eigenvectors

if has distinct roots, then the eigenvectors form a basis

Eigenvalues of are , let be respectively the corresponding eigenvectors

The eigenvalue of is (with multiplicity ), let be an eigenvector of it

the dimension of the eigenspace is

find eigenvectors of following matrices

let be eigenvalue of , and the characteristic polynomial

（the trace of ）

takes eigenpairs and

The choice of and in is not unique

example (permuting columns in ):

The choice of and in is not unique

example (scaling columns in ):

if can be diagonalized as , then

if is a polynomial, then

Cayley-Hamilton Theorem： if is the characteristic polynomial of , then

if are diagonal matrices, then clearly

if can be diagonalized by the same , i.e., ,, then

simultaneously diagonalizable multiplicative commutativity

concepts and computations

product and sum of eigenvalues

diagonalization