linear algebra review 061904 -...

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


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


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


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


Matrix Notation

Square

[ ]

=

5554535251

4544434241

3534333231

2524232221

1514131211

aaaaaaaaaaaaaaaaaaaaaaaaa

A

[ ]

=

5545352515

4544342414

3534332313

2524232212

1514131211


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


A [ ]

=

244

233

222

211

aa1aa1aa1aa1

A

A matrix [A] can have some special forms


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=



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


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



Multiplication Rules]A][B[]C[]B][A[ ≠=

]C][A[]B][A[])C[]B]([A[ +=+

])C][B]([A[]C])[B][A([ =



Pre-Multiplication by a Diagonal Matrix

=

nm2n1nnn

im2i1iii

m2222122

m1121111

aaad

aaad

aaadaaad

]A[D

L

L

L

L

O

O



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



Transpose of a Matrix

==⇒

=ji

2212

2111

T

ij

2221

1211

a

aaaa

]A[]B[a

aaaa

]A[



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[ ===



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([ −−− =


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 =


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


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



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



The individual terms of the decomposition using a process such as Crout gives

rjirijijjj

jirijijriiriiii ulau;

uula

l;ulau ∑∑∑ −=−

=−=



The simple 3x3 stiffness matrix can be decomposed to be

−

−

−−=

−−−

−

0.115.1

012

1667.00.015.0

1

11121

12



Each of the factors are given by

[ ] [ ][ ]

( )

−−−−

=

∑∑∑∑

133133221331321131

1331231221221121

131211

aaaaaaaaaaaaaaaaa

aaa

ULA



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



⇓↓⇓↓⇓↓⇓↓

⇒⇒⇒⇒⇒⇓↓→→→→→→↓

⇓↓⇓↓⇓↓⇓↓⇓↓

→→→→→→↓

↓↓↓↓↓

→→→→→→↓

↓↓↓↓↓↓

nn14131211nn14131211

nn14131211nn14131211

aaaaa

then

aaaaa

aaaaa

then

aaaaa


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−



Define an Eigenproblem

Generalized Inverse

[ ] [ ][ ]{ } { } { }i2i x;0XBA ω⇒=λ−

{ } [ ]{ } { } [ ] { }xUppUx g=⇒=

[ ] [ ] [ ]( ) [ ]T1Tg UUUU−

=



Moore-Penrose Conditions for Generalized Inverse

[ ][ ] [ ] [ ]UUUU g =

[ ] [ ][ ] [ ]ggg UUUU =

[ ] [ ]( ) [ ] [ ]UUUU gTg =

[ ][ ]( ) [ ][ ]gTg UUUU =


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



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



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



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



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 +=

=



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

+−

−+

=


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



Finite element model development uses individual elements that are assembled into system matrices



Structural system equations - coupled

Eigensolution - eigenvalues & eigenvectors

[ ]{ } [ ]{ } [ ]{ } { })t(FxKxCxM =++ &&&

[ ] [ ][ ]{ } 0xMK =λ−

{ } { } { } [ ] { }FUp\

K\

p\

C\

p\

M\

T=

+

+

&&&

Modal space representation of equations - uncoupled



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



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



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)



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

−+

−= ∑

=



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



Many other applications exist

Correlation Model UpdatingAdvanced Data Manipulation

Operating Data Rotating EquipmentNonlinearities Modal Parameter Estimation

and the list goes on and on

linear algebra review 061904 -...

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