lecture 9 symmetric matrices subspaces and nullspaces
DESCRIPTION
Lecture 9 Symmetric Matrices Subspaces and Nullspaces. Shang-Hua Teng. Matrix Transpose. Addition: A+B Multiplication: AB Inverse: A -1 Transpose : A -T. Transpose. Inner Product and Outer Product. Properties of Transpose. End of Page 109: for a transparent proof. 0. R. r. - PowerPoint PPT PresentationTRANSCRIPT
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Lecture 9Symmetric Matrices
Subspaces and Nullspaces
Shang-Hua Teng
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Matrix Transpose
• Addition: A+B
• Multiplication: AB
• Inverse: A-1
• Transpose : A-T
jiij AA
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Transpose
84
73
62
51
8765
4321T
5
4
3
2
1
54321 T
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Inner Product and Outer Product
703221125
8
7
6
5
4321
3224168
2821147
2418126
2015105
4321
8
7
6
5
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Properties of Transpose
11
TT
TTT
TTT
AA
ABAB
BABA
End of Page 109: for a transparent proof
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Ellipses and Ellipsoids
122
r
y
R
x
1/10
0/1,
y
x
r
Ryx
R
r
0
1
/10
00
0/1 11
1
nn
n
x
x
r
r
xx
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Later
R
r 0
yAxyAx TTT
Relating to
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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
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Symmetric Matrix
n
TTT
d
d
D
BDBBBBB
1
, ,
B is an m by n matrix
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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
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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
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What Vector Spaces Do We Know So Far
• Rn: the space consists of all column (row) vectors with n components
nRRRR ,,,, 321
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Properties of Vector Spaces
xxxxxx
cycxyxcbyaxxba
bxaxabzyxzyx
xyyx
00)(
0x0 ;1 ;)(
)( );()(
)()( ;
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Other Vector Spaces
matrices by real all ofset the:M
: functions real all ofset the:F
0 :Z
nm
RRxf
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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
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General Linear System
The system Ax =b is solvable if and only if b is in C(A)
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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
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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.
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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
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Subspace of 2 by 2 matrices
a
ad
a
d
ba
0
0 :I of multiple all ofset the
0
0 matrices diagonal all ofset the:D
0 matricesngular upper tria all ofset the:U
00
00 :Z
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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)}
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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
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Permute Rows and Continuing Elimination (permute columns)
011121
131111
021111
110011
A
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Theorem
If Ax = 0 has more have more unknown than equations (m > n: more columns than rows), then it has nonzero solutions.
There must be free variables.
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Echelon Matrices
*
*
*
*
000000
**0000
*****0
******
A
Free variables