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- 5주차 강의 내용 -
8.3 Symmetric, Skew-Symmetric, and Orthogonal Matrices
DefinitionsA real square matrix A = [ajk] is called
symmetric if transposition leaves it unchanged,
(1) AT = A, thus
skew-symmetric if transposition gives the negatives of A,
(2) AT = -A, thus
orthogonal if transposition gives the inverse of A,
(3) AT = A-1,
jk
,
kj
a
a
jk
,
kj
a
a
Ex. 1) Symmetric, Skew-Symmetric, and Orthogonal Matrices The matrices below are symmetric, skew-symmetric and orthogonal,
respectively.
3 2 3 2 3
1 3
1 3 2 3 2
3 2 3 1 3 2
, 0 20 12
20 0
9
12 9
0 , 4 2 5
2 0 1
5 1 3
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
Ex. 2) Symmetric, Skew-Symmetric, and Orthogonal Matrices
3 2 3 1 3
2 3
1 3 2 3
2 3
2 3 2 4
1
, 0 12 8
12 0 4
8 4 0 , 8 1 4
1 8 4
4 4 7
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
Any real square matrix A may be written as the sum of a symmetric matrix R and a skew-symmetric matrix S, where
(4) ( )
2 ) 1
2 (
1 T
A A
A A S
R T
Ex. 3) Illustration of Formula (4)
0 0 . 6 5 . 1
0 . 6 0 5 . 1
5 . 1 5 . 1 0
0 . 3 0 . 2 5 . 3
0 . 2 0 . 3 5 . 3
5 . 3 5 . 3 0 . 9
3 4 5
8 3 2
2 5 9
S R A
Theorem 1
Eigenvalues of Symmetric and Skew-Symmetric Matrices
(a) The eigenvalues of a symmetric matrix are real.
(b) The eigenvalues of a skew-symmetric matrix are pure imaginary or zero.
A=
3 1
1
3 det(A-λI)= (3 ) 1 0, 4, 2
3 1
1
3 2
Ex)
☞ The eigenvalues of a symmetric matrix A are real.
1 0 1
0 det(B-λI) = i
1 0,
1
1 2
Ex) B=
☞ The eigenvalues of a skew-symmetric matrix A are pure imaginary.
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
C= det(C-λI)=
Ex)
8 1 4
1 8 4
4 4 7
0 ) 9 )(
9 ( 8
1 4
1 8
4
4 4
7
2
λ λ λ λ
λ
9 , 9
λ
☞ The eigenvalues of a symmetric matrix A are real.
Theorem 4
Determinant of an Orthogonal Matrix
The determinant of an orthogonal matrix are the value +1 or -1.
Theorem 5
Eigenvalues of an Orthogonal Matrix
The eigenvalues of an orthogonal matrix
Aare real or complex conjugates in pairs and have absolute value
1.
3 2 3 2 3 1
3 1 3 2 3 2
3 2 3 1 3
2 det(C-λI)
1 0
3 2 3
2
23
Ex) C =
, 0 3 2 2
3
3
2
6 11 , 5
1 i
1 ) 6 / 11 ( ) 6 / 5
( 2 2
☞ The eigenvalues of an orthogonal matrix C are real or complex conjugates in pairs and have absolute value 1.
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
8.4 Similarity, Diagonalization
Definition
Similar Matrices, Similarity Transformation
An n x n matrix  is called similar to n x n matrix A if (4)
for some (nonsingular) n x n matrix P. This information, which gives  from A, is called a similarity transformation.
AP P
A
1Theorem 3
Eigenvalues and Eigenvectors of Similar Matrices
If
Âis similar to
A, then
Âhas the same eigenvalues as
A. Furthermore, if
xis an eigenvector of
A, then
y = P-1xis an eigenvector of
Âcorres- ponding to the same eigenvalue.
Ex. 4) Eigenvalues and Vectors of Similar Matrices
1 4
3
A 6
and
4 1
3 P 1
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
2 0
0 3 4 1
3 1 1 4
3 6 1 1
3 A 4
1 4
3
A 6 det(A-λI) = 5 6 12 0, 3,2
1 4
3
6 2
From
2 0
0 ˆ 3
A det( -λI) = ( 3)( 2) 0, 3,2
2 0
0
3
From Aˆ
☞ Similar Matrices have the same eigenvalues.
When
4
, 3 3 4
; 3 0 4
3 2 4
;
2 A I x1 x2 X1
1 , , 0
0
; 0 0 0
0 2 1
ˆ 1 1
I x Y
A 1 1 1
1 0 4 3 1 1
3
4 Y
X
P
When
1
, 1
; 4 0 4
3 3 3
;
3 A I x1 x2 X2
0 , , 1
0
; 1 0 0
0 3 0
ˆ 1 2
I y Y
A 2 2
1
0 1 1 1 1 1
3
4 Y
X
P
☞ y = P-1x is an eigenvector of Â
☞ y = P-1x is an eigenvector of  eigenvector of Â
eigenvector of Â
eigenvector of A
eigenvector of A
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
Theorem 4
Diagonalization of a Matrix
If an n x n matrix
Ahas a basis of eigenvectors, then
(5) D = X-1AX
is diagonal, with the eigenvalues of
Aas the entries on the main diagonal.
Here
Xis the matrix with these eigenvectors as column vectors.
Ex. 5) Diagonalization
10 6
9
A 5 Characteristic equation ;
,
11
with λ (A xI) 0. Its eigenvector is
1 2
1 X 3
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
det(A-λI) = 5 4 0, 1,4
10 6
9
5 2
λ λ λ
λ λ
0 9 6 1 2
x x
2 3 X1
,
14
with λ (A4I)x0. 9x19x2 0 Its eigenvector is
1 1 X2
3 2
1
1 1
X
4 0
0 1 1 2
1 3 10 6
9 5 3 2
1
1AX 1 X D
4 3 1
1 1 3
2 1 1
3 . 9 8 . 1 7 . 17
5 . 5 0 . 1 5 . 11
7 . 3 2 . 0 3 . 7
2 . 0 2 . 0 8 . 0
7 . 0 2 . 0 3 . 1
3 . 0 2 . 0 7 . 0
1AX X D
0 0 0
0 4 0
0 0 3
0 12 3
0 4 9
0 4 3
2 . 0 2 . 0 8 . 0
7 . 0 2 . 0 3 . 1
3 . 0 2 . 0 7 . 0
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
Ex. 5) Diagonalization
5 . 5 0 . 1 7 . 17
5 . 5 0 . 1 5 . 11
7 . 3 2 . 0 3 . 7 A
Characteristic equation ;
3
2 12 0 , 0 ,
4 ,
3
2 31
,
1
3
with
(
A
1I)
x 0 .
Its eigenvector is
1 3 1
T,
2
4
with
(
A
2I)
x 0 .
Eigenvector is
1 1 3
T,
3
0
with
(
A
3I)
x 0 .
Eigenvector is
2 1 4
T
4 3 1
1 1 3
2 1 1 X
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
Ex. 6) Diagonalization
1 2 1
0 1 6
1 2 1 A
, 0 ) 3 )(
4 ( , 0 12 )
det(AλI λ3λ2 λ λ λ λ 3 ,
4 ,
0 2 3
1
From Ex. 3) 8.1
13 6 1 X1
1 2 1 X2
2 3 2 X3
2 1 13
3 2 6
2 1 1 X
21 / 2 7 / 1 21 / 9
28 / 3 7 / 2 28 / 9
12 / 1 0 12 / 1 X 1
21 / 2 7 / 1 21 / 9
28 / 3 7 / 2 28 / 9
12 / 1 0 12 / 1
1AX X D
1 2 1
0 1 6
1 2 1
2 1 13
3 2 6
2 1 1
3 0 0
0 4 0
0 0 0
8.5 Complex Matrices and Forms Notations
is obtained from by replacing each entry ( ; real) with its complex conjugate . Also, is the transpose of , hence the conjugate transpose of A.
] [ajk
A A[ajk] ,
i
ajk AT[akj]
i ajk
A
Definition
Hermitian, Skew-Hermitian, and Unitary Matrices A square matrix A = is called
Hermitian if , that is ,
skew-Hermitian if , that is ,
unitary if
] [ a
jkA
A T akj
ajk AA T
a
kj a
jk1
A A T
Ex. 2) Hermitian, Skew-Hermitian, and Unitary Matrices
i i
i C i
i B i
i A i
2 3 1 2 1
2 3 1 2
1 2 ,
2 , 3
7 3
1
3 1 4
are Hermitian, skew-Hermitian, and unitary matrices.
Chap. 8 Linear Algebra : Matrix Eigenvalue Problems
Ex. 1) Notations
If ,
5 2 6
1 4
3
i i
A i then ,
5 2 6
1 4
3
i i
A i and
i i
AT i
5 2 1
6 4 3
- Theorem 1 Eigenvalues
(a) The eigenvalues of a Hermitian matrix are real.
* symmetric matrix(b) The eigenvalues of a skew-Hermitian matrix are pure imaginary or zero.
(c) The eigenvalues of a unitary matrix have absolute value
1.
* orthogonal matrix Ex. 3) Illustration of Theorem 1
From the matrices in Example 2,
Matrix Characteristic Equation Eigenvalues
Hermitian 9, 2
Skew-Hermitian Unitary
0 18
2 11
0 8
2
2
i0
2
1
ii i, 2 4
i
i 2
3 1 2 , 1 2 3 1 2
1
* skew-symmetric matrix
