Linear Algebra review 061904 -...
Transcript of Linear Algebra review 061904 -...
![Page 1: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/1.jpg)
1 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Review Linear Algebra
Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell
[ ] [ ][ ][ ]ADet
AintAdjoA 1=−
[ ]
=
x0..0.x0....x0....x0x...x
U
[ ] { } { }[ ] [ ] [ ] [ ]{ } [ ] { } [ ] [ ] [ ][ ] { }{ } [ ] [ ] [ ][ ]{ }n
Tnn
gnmmmm
ngT
mmnmnnngnmm
Tmmnmnnnm
nmnm
BVSUX
BUSVBAX
USVA
BXA
=
==
=
=[ ] { } { } { }[ ]
{ }{ }{ }
=
MO
L T3
T2
T1
3
2
1
321 vvv
ss
s
uuuA
[ K ]n
[ M ]n [ M ]a[ K ]a [ E ]a
[ ω ]2
Structural Dynamic Modeling Techniques & Modal Analysis Methods
![Page 2: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/2.jpg)
2 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra
The analytical treatment of structural dynamic systems naturally results in algebraic equations that are best suited to be represented through the use of matrices
Some common matrix representations and linear algebra concepts are presented in this section
![Page 3: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/3.jpg)
3 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra
Common analytical and experimental equations needing linear algebra techniques
[ ] [ ] [ ]ffyf GHG =
[ ]{ } [ ]{ } [ ]{ } { })t(FxKxCxM =++ &&& [ ] [ ][ ]{ } 0xMK =λ−
[ ] [ ][ ] 1ffyf GGH −=
( )[ ] ( ){ } ( ){ }sFsxsB = ( )[ ] ( )[ ] ( )[ ]( )[ ]sBdetsBAdjsHsB 1 ==−
( )[ ] [ ] [ ]TLSUsH
=
O
O
or
![Page 4: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/4.jpg)
4 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Matrix Notation
A matrix [A] can be described using row,column as
[ ]
=
54535251
44434241
34333231
24232221
14131211
aaaaaaaaaaaaaaaaaaaa
A
( row , column )
[A]T -Transpose - interchange rows & columns[A]H - Hermitian - conjugate transpose
![Page 5: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/5.jpg)
5 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Matrix Notation
Square
[ ]
=
5554535251
4544434241
3534333231
2524232221
1514131211
aaaaaaaaaaaaaaaaaaaaaaaaa
A
[ ]
=
5545352515
4544342414
3534332313
2524232212
1514131211
aaaaaaaaaaaaaaaaaaaaaaaaa
A
[ ]
=
55
44
33
22
11
aa
aa
a
A [ ]
=
55
4544
353433
25242322
1514131211
a0000aa000aaa00aaaa0aaaaa
A
Triangular Diagonal
Symmetric Vandermonde Toeplitz
[ ]
=
54321
65432
76543
87654
98765
aaaaaaaaaaaaaaaaaaaaaaaaa
A [ ]
=
244
233
222
211
aa1aa1aa1aa1
A
A matrix [A] can have some special forms
![Page 6: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/6.jpg)
6 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
General Matrix
=
nm1n1n
ij
m12221
m11211
aaa
a
aaaaaa
]A[
L
MMM
L
L
L Column Vector
Row Vector
=
n
i
2
1
b
b
bb
}B{
M
M
mj21 ccccC LL=
![Page 7: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/7.jpg)
7 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
AdditionScalar MultiplierMatrix Multiplication
ijijij bac]B[]A[]C[ +=⇒+=
ijij a*sb]A[*s]B[ =⇒=
{ }jiij bac]B][A[]C[ =⇒=
∑=⇒
=k
kjikij
kj
j2
j1
ik2i1iij bac
b
bb
aaac
ML
![Page 8: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/8.jpg)
8 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Matrix Manipulation
A matrix [C] can be computed from [A] & [B] as
=
3231
2221
1211
5251
4241
3231
2221
1211
3534333231
2524232221
1514131211
cccccc
bbbbbbbbbb
aaaaaaaaaaaaaaa
5125412431232122112121 bababababac ++++=
∑=k
kjikij bac
![Page 9: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/9.jpg)
9 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
Multiplication Rules]A][B[]C[]B][A[ ≠=
]C][A[]B][A[])C[]B]([A[ +=+
])C][B]([A[]C])[B][A([ =
![Page 10: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/10.jpg)
10 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
Pre-Multiplication by a Diagonal Matrix
=
nm2n1nnn
im2i1iii
m2222122
m1121111
aaad
aaad
aaadaaad
]A[D
L
L
L
L
O
O
![Page 11: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/11.jpg)
11 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
Post-Multiplication by a Diagonal Matrix
=
ni
i2
i1
ii
2n
22
12
22
1n
21
11
11
a
aa
d
a
aa
d
a
aa
dD]A[MMM
O
O
![Page 12: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/12.jpg)
12 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
Transpose of a Matrix
==⇒
=ji
2212
2111
T
ij
2221
1211
a
aaaa
]A[]B[a
aaaa
]A[
![Page 13: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/13.jpg)
13 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
Transposition RulesTTT ]B[]A[])B[]A([ +=+
[ ][ ] ]A[ATT = TTT ]A[]B[])B][A([ =
( ) TTTT ]A[]B[]C[]C][B][A[ =
Symmetric Rules
( )TTT ]B][A[]B][A[;]B[]B[;]A[]A[ ≠==
[ ] [ ]TTT C]C[;]B][A[]B[]C[;A]A[ ===
![Page 14: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/14.jpg)
14 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
Inverse of a Matrix
Properties of an Inverse
[ ]ij)ji(
ijT1 M)1(cwhere]C[]A[Adj;
]Adet[]A[Adj]A[ +− −===
[ ][ ] ]A[A11 =
−−
111 ]A[]B[])B][A([ −−− =
![Page 15: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/15.jpg)
15 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Simple Set of Equations
A common form of a set of equations is
Underdetermined # rows < # columnsmore unknowns than equations(optimization solution)
Determined # rows = # columnsequal number of rows and columns
Overdetermined # rows > # columnsmore equations than unknowns(least squares or generalized inverse solution)
[ ]{ } [ ]bxA =
![Page 16: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/16.jpg)
16 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Simple Set of Equations
3zy2z1y2x
1yx2
=+−=−+−
=−
=
−−−
−
321
zyx
110121
012
This set of equations has a unique solution
whereas this set of equations does not
2y2x42z1y2x
1yx2
=−=−+−
=−
=
−−−
−
221
zyx
024121
012
![Page 17: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/17.jpg)
17 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Static Decomposition
A matrix [A] can be decomposed and written as
Where [L] and [U] are the lower and upper diagonal matrices that make up the matrix [A]
[ ] [ ] [ ]ULA =
[ ] [ ]
=
=
x0000xx000xxx00xxxx0xxxxx
U
xxxxx0xxxx00xxx000xx0000x
L
![Page 18: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/18.jpg)
18 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Static Decomposition
Once the matrix [A] is written in this form then the solution for {x} can easily be obtained as
[ ]{ } [ ] [ ]BLXU 1−=
[ ] [ ] [ ]ULA =
Applications for static decomposition and inverse of a matrix are plentiful. Common methods are
Gaussian elimination Crout reductionGauss-Doolittle reduction Cholesky reduction
![Page 19: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/19.jpg)
19 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Static Decomposition
The individual terms of the decomposition using a process such as Crout gives
rjirijijjj
jirijijriiriiii ulau;
uula
l;ulau ∑∑∑ −=−
=−=
![Page 20: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/20.jpg)
20 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Static Decomposition
The simple 3x3 stiffness matrix can be decomposed to be
−
−
−−=
−−−
−
0.115.1
012
1667.00.015.0
1
11121
12
![Page 21: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/21.jpg)
21 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Static Decomposition
Each of the factors are given by
[ ] [ ][ ]
( )
−−−−
=
∑∑∑∑
133133221331321131
1331231221221121
131211
aaaaaaaaaaaaaaaaa
aaa
ULA
![Page 22: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/22.jpg)
22 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Static Decomposition
Each of the equations are processed in the order shown below where the first row is retained followed by the decomposition of the first column followed by the decomposition of the 2nd row starting from the 2-2 position and so on until the entire matrix is decomposed.
652431
aaa 131211
![Page 23: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/23.jpg)
23 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Static Decomposition
⇓↓⇓↓⇓↓⇓↓
⇒⇒⇒⇒⇒⇓↓→→→→→→↓
⇓↓⇓↓⇓↓⇓↓⇓↓
→→→→→→↓
↓↓↓↓↓
→→→→→→↓
↓↓↓↓↓↓
nn14131211nn14131211
nn14131211nn14131211
aaaaa
then
aaaaa
aaaaa
then
aaaaa
![Page 24: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/24.jpg)
24 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Eigenvalue Problems
Many problems require that two matrices [A] & [B] need to be reduced
Applications for solution of eigenproblems are plentiful. Common methods are
Jacobi Givens HouseholderSubspace Iteration Lanczos
[ ]{ } [ ]{ } { })t(QxBxA =+&& [ ] [ ][ ]{ } 0xAB =λ−
![Page 25: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/25.jpg)
25 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
Define an Eigenproblem
Generalized Inverse
[ ] [ ][ ]{ } { } { }i2i x;0XBA ω⇒=λ−
{ } [ ]{ } { } [ ] { }xUppUx g=⇒=
[ ] [ ] [ ]( ) [ ]T1Tg UUUU−
=
![Page 26: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/26.jpg)
26 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Rules & Definitions
Moore-Penrose Conditions for Generalized Inverse
[ ][ ] [ ] [ ]UUUU g =
[ ] [ ][ ] [ ]ggg UUUU =
[ ] [ ]( ) [ ] [ ]UUUU gTg =
[ ][ ]( ) [ ][ ]gTg UUUU =
![Page 27: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/27.jpg)
27 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Singular Valued Decomposition
[ ] [ ][ ][ ]TVSUA =
Any matrix can be decomposed using SVD
[U] - matrix containing left hand eigenvectors[S] - diagonal matrix of singular values[V] - matrix containing right hand eigenvectors
![Page 28: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/28.jpg)
28 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Singular Valued Decomposition
SVD allows this equation to be written as
which implies that the matrix [A] can be written in terms of linearly independent pieces which form the matrix [A]
[ ] { } { } { }[ ]
{ }{ }{ }
=
MO
L T3
T2
T1
3
2
1
321 vvv
ss
s
uuuA
[ ] { } { } { } { } { } { } L+++= T333
T222
T111 vsuvsuvsuA
![Page 29: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/29.jpg)
29 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Singular Valued Decomposition
Assume a vector and singular value to be
1sand321
u 11 =
=
Then the matrix [A1] can be formed to be
[ ] { } { } [ ]{ }
=
==
963642321
3211321
usuA T1111
The size of matrix [A1] is (3x3) but its rank is 1There is only one linearly independent
piece of information in the matrix
![Page 30: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/30.jpg)
30 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Singular Valued Decomposition
Consider another vector and singular value to be
1sand1
11
u 22 =
−=
Then the matrix [A2] can be formed to be
[ ] { } { } [ ]{ }
−−−−
=−
−==
111111111
11111
11
usuA T2222
The size and rank are the same as previous caseClearly the rows and columns
are linearly related
![Page 31: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/31.jpg)
31 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Singular Valued Decomposition
Now consider a general matrix [A3] to be
The characteristics of this matrix are not obvious at first glance.
Singular valued decomposition can be used to determine the characteristics of this matrix
[ ] [ ] [ ]213 AA1052553232
A +=
=
![Page 32: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/32.jpg)
32 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Singular Valued Decomposition
The SVD of matrix [A3] is
or
[ ]{ }{ }{ }
−
−
=
000111321
01
1
000
111
321
A
These are the independent quantities that make up the matrix which has a rank of 2
[ ] { } { } { }TTT 0000000
11111
11
3211321
A
+−
−+
=
![Page 33: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/33.jpg)
33 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Applications
The basic solid mechanics formulations as well as the individual elements used to generate a finite element model are described by matrices
L
E, I
F F
θ i
i j
θ j
ν i ν j
{ } [ ]{ }
γγγεεε
=
τττσσσ
⇒ε=σ
yz
xz
xy
z
y
x
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
yz
xz
xy
z
y
x
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
[ ]
−−−−
−−
=
L4L6L2L6L612L612
L2L6L4L6L612L612
LEIk
2
22
3
[ ]
−−
−−−−
ρ=
22
22
L4L22L3L13L22156L1354L3L13L4L22L1354L22156
420ALm
![Page 34: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/34.jpg)
34 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Applications
Finite element model development uses individual elements that are assembled into system matrices
![Page 35: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/35.jpg)
35 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Applications
Structural system equations - coupled
Eigensolution - eigenvalues & eigenvectors
[ ]{ } [ ]{ } [ ]{ } { })t(FxKxCxM =++ &&&
[ ] [ ][ ]{ } 0xMK =λ−
{ } { } { } [ ] { }FUp\
K\
p\
C\
p\
M\
T=
+
+
&&&
Modal space representation of equations - uncoupled
![Page 36: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/36.jpg)
36 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Applications
Multiple Input Multiple Output Data Reduction
FREQUENCY RESPONSE FUNCTIONS FORCE
[H] [Gxx][Gyx]
RESPONSE
=
=
(MEASURED) (UNKNOWN) (MEASURED)
[ ] [ ] [ ]xxyx GHG = [ ] [ ][ ] 1xxyx GGH −=
Matrix inversion can only be performed if the matrix [Gxx] has linearly independent inputs
![Page 37: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/37.jpg)
37 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Applications
Principal Component Analysis using SVD
[Gxx]
SVD of the input excitation matrix identifies the rank of the matrix - that is an indication of how many linearly independent inputs exist
[ ] { } { } { }[ ]
{ }{ }{ }
=
MO
L T
T2
T1
2
1
21xx 0vv
0s
s
0uuG
![Page 38: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/38.jpg)
38 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Applications
SVD of Multiple Reference FRF Data
SVD of the [H] matrix gives an indication of how many modes exist in the data
[ ] { } { } { }[ ]
{ }{ }{ }
=
MO
L T3
T2
T1
3
2
1
321 vvv
ss
s
uuuH
FREQUENCY RESPONSE FUNCTIONS
[H]
0 50 100 150 200 250 300 350 400 450 500Frequency (Hz)
![Page 39: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/39.jpg)
39 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Applications
Least Squares or Generalized Inverse for Modal Parameter Estimation Techniques
Least squares error minimization of measured data to an analytical function
( )[ ] [ ]( )
[ ]( )*
k
*k
j
ik k
k
ssA
ssAsH
−+
−= ∑
=
![Page 40: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/40.jpg)
40 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Applications
Extended analysis and evaluation of systems
[ ][ ] [ ][ ][ ]2I `UMUK ω=
[ ][ ][ ] [ ] [ ][ ][ ]2I
TI
T `UMUUKU ω=
[ ] [ ] [ ] [ ][ ][ ][ ][ ] [ ][ ] [ ][ ][ ] [ ][ ]TI
TSI
TS
S2T
SI
MUUKMUUK
VK`VKK
−−
+ω+=
[ ] [ ] [ ] [ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ]TSSS
2TSI VUKVUKVK`VKK −−+ω+=
generally require matrix manipulation of some type
![Page 41: Linear Algebra review 061904 - uml.edufaculty.uml.edu/.../22.515/Linear_Algebra_review_061904.pdf · 2004-06-29 · 22.515 - Linear Algebra Concepts Static Decomposition Each of the](https://reader034.fdocuments.us/reader034/viewer/2022042221/5ec76cb5767a1b76f573b39f/html5/thumbnails/41.jpg)
41 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 - Linear Algebra Concepts
Linear Algebra Applications
Many other applications exist
Correlation Model UpdatingAdvanced Data Manipulation
Operating Data Rotating EquipmentNonlinearities Modal Parameter Estimation
and the list goes on and on