Def of Eigenvalue and Eigenvector:
Let
be a n-by-n matrix. A scalar is called an eigenvalue of if there exist a nonzero vector such that The vector
is called an eigenvector of corresponding to the eigenvalue
Theorem(找Eigenvalue的方法):
If
is an n-by-n matrix, then is an eigenvalue of if and only if
Proof:
is an eigenvalue of
such that
has nontrivial solutions
is not invertible
Def of eigenspace(所有eigenvector和0向量組成的向量空間):
let
be the set of zero vector and all eigenvectors of an n-by-n matrix corresponding the eigenvalue .
Theorem(
The eigenspace of an n-by-n matrix
corresponding to the eigenvalue i.e. is a subspace of R^n.
Proof:
Let
and is a scalar
(closure)
(closure)
is a subspace of
Def of Matrix similarity:
If
and are square matrices of the same size, then we say that is simliar to if there exist an invertible matrix such that Similar matrices have many properties in common. If
and are simliar matrices, then they have the same determinant and eigenvalues.
In general, only property that is shared by all similar matrices is called a similariry invariant or is said to be invariant under similarity.
Def of diagonalizable square matrix:
A square matrix
is said to be diagonalizable if there exist an invertible matrix such that is diagonal. In this case, the invertible matrix is said to be diagonalize .
Theorem(判斷方陣是否可對角線化):
If
is an n-by-n matrix then is diagonalizable if and only if has n linearly independent eigenvectors.
is diagonalizable
has n linearily independent eigenvectors, say,
where are scalars and need not all distinct. Let
then
Procedure for diagonalization on n-by-n matrix A:
- Find all eigenvelues
of . - Find all eigenvectors X of
- Number of independent eigenvectors of
or No: is not diagonalizable. Yes: let be the linearly independent eigenvectors. - Let
then diagonalizes such that
對角線化與可逆之間沒有必然聯繫
Find power of a matrix ：
對於可對角線化的矩陣A有:
注意：
即對主軸元素做n次平方
Theorem(存在固有值為0的方陣必不可逆，反之亦然):
A square matrix
is singular if and only if is an eigenvalue of A.
Proof:
is an eigenvalue of
A is singular.