Lecture 9 Symmetric Matrices Subspaces and Nullspaces Shang-Hua Teng
Dec 20, 2015
Lecture 9Symmetric Matrices
Subspaces and Nullspaces
Shang-Hua Teng
Matrix Transpose
• Addition: A+B
• Multiplication: AB
• Inverse: A-1
• Transpose : AT
jiijT AA
Transpose
84
73
62
51
8765
4321T
5
4
3
2
1
54321 T
Inner Product and Outer Product
703221125
8
7
6
5
4321
3224168
2821147
2418126
2015105
4321
8
7
6
5
Properties of Transpose
11
TT
TTT
TTT
AA
ABAB
BABA
End of Page 109: for a transparent proof
Ellipses and Ellipsoids
122
r
y
R
x
1/10
0/1,
2
2
y
x
r
Ryx
R
r
0
1
/10
00
0/1 1
2
21
1
nn
n
x
x
r
r
xx
Later
R
r 0
yAxyAx TTT
Relating to
Symmetric Matrix
• Symmetric Matrix: A= AT
1;John2:Alice
4:Anu3:Feng
Graph of who is friend with whomand its matrix
1101
1110
0111
1011
Examples of Symmetric Matrices
n
TTT
d
d
D
BDBBBBB
1
, ,
B is an m by n matrix
Elimination on Symmetric Matrices
• If A = AT can be factored into LDU with no row exchange, then U = LT. In other words
The symmetric factorization of a symmetric matrix is A = LDLT
10
21
40
01
12
01
82
21
So we know Everything about Solving a Linear System
• Not quite but Almost
• Need to deal with degeneracy (e.g., when A is singular)
• Let us examine a bigger issues:
Vector Spaces and Subspaces
What Vector Spaces Do We Know So Far
• Rn: the space consists of all column (row) vectors with n components
nRRRR ,,,, 321
Properties of Vector Spaces
xxxxxx
cycxyxcbyaxxba
bxaxabzyxzyx
xyyx
00)(
0x0 ;1 ;)(
)( );()(
)()( ;
Other Vector Spaces
matrices by real all ofset the:M
: functions real all ofset the:F
0 :Z
nm
RRxf
Vector Spaces Defined by a Matrix
nRxAxAC :)(
For any m by n matrix A
• Column Space:
• Null Space: 0:)( AxxAN
222
111
010
01
N
N
General Linear System
The system Ax =b is solvable if and only if b is in C(A)
Subspaces
• A subspace of a vector space is a set of vectors (including 0) that satisfies two requirements: if v and w are vectors in the subspace and c is any scalar, then– v+w is in the subspace– cv is in the subspace
Subspace of R3
• (Z): {(0,0,0)}
• (L): any line through (0,0,0)
• (P): any plane through (0,0,0)
• (R3) the whole space
A subspace containing v and w must contain all linear combination cv+dw.
Subspace of Rn
• (Z): {(0,0,…,0)}• (L): any line through (0,0,…,0)• (P): any plane through (0,0,…,0)• …• (k-subspace): linear combination of any k independent
vectors • (Rn) the whole space
Subspace of 2 by 2 matrices
a
a
d
a
d
ba
0
0 :I of multiple all ofset the
0
0 matrices diagonal :D
0 matricesngular upper tria :U
00
00 :Z
Express Null Space by Linear Combination
• A = [1 1 –2]: x + y -2z = 0
x = -y +2z
Free variablesPivot variable
• Set free variables to typical values
(1,0),(0,1)
• Solve for pivot variable: (-1,1,0),(2,0,1)
{a(-1,1,0)+b(2,0,1)}
Express Null Space by Linear Combination
2032
3121A
Guassian Elimination for finding the linear combination: find an elimination matrix E such that
EA = free
pivot
Permute Rows and Continuing Elimination (permute columns)
011121
131111
021111
110011
A
Theorem
If Ax = 0 has more unknown than equations (m > n: more columns than rows), then it has nonzero solutions.
There must be free variables.
Echelon Matrices
*
*
*
*
000000
**0000
*****0
******
A
Free variables