Linear Transformation Examples Matrix Eigenvalue problems

Md67
Md68
Similarity of Matrices
● Eigenvectors and their properties:
The eigenvectors of an (n × n) matrix A may or may not form a
basis for Rn . If they do (e.g. case of n distinct eigenvalues), then
they can be used for “diagonalising” A - i.e. transforming A into
diagonal form with the eigenvalues on the main diagonal.
● Similarity transformation:
n× n A√ n× n A A√ −1
matrix is similar to matrix if = P AP
for some nonsingular n× n matrix P.
Then A√ has the same eigenvalues as A, and if x is an eigenvector of A,
−1
then P x is an eigenvector of A√
corresponding to the same eigenvalue.
● Basis of eigenvectors:
If λ1, λ2, ..., λk,
are distinct eigenvalues of a matrix, then the corresponding
eigenvectors x1, x2, ..., xkform a linearly independent set. If n× n matrix
A has n distinct eigenvalues, then A
n
has a basis of eigenvectors for R .
● Basis of eigenvectors
Diagonalization
A symmetric matrix always has an orthonormal basis of eigenvectors for
e.g. has orthonormal basis of eigenvectors 1
n
R .
A = ,
2
1
;
2
corresponding to eigenvalues , and respectively.
⎡
⎢
⎣
⎤
⎥
⎦
⎡
⎢
⎣
⎤
5
3
3
5
1
1⎥
⎦
⎡ 1 ⎤
⎢
⎣−1⎥
⎦
λ = 8 λ = 2
● Diagonalization of a matrix
1
If n× n matrix A has a basis of eigenvectors, then D= X AX is diagonal,
with the eigenvalues of A as the entries on the main diagonal,
and where X is the matrix with these eigenvectors as column vectors.
m m
Also D = X A X m =
e.g. A= has X and X giving X AX
⎡
−1
−1
, 2, 3,
...
5
⎢
⎣1
4⎤
⎡4
⎥ , =
2 ⎢
⎦ ⎣1
1 ⎤
⎡−
−1 1 1
− ⎥ , =
1⎦
−5
⎢
⎣−1
−1⎤
−1
⎡6
⎥ ,
=
4
⎢
⎦
⎣0
0⎤
1⎥
⎦
2