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

[ ] { } { }[ ] [ ] [ ] [ ]{ } [ ] { } [ ] [ ] [ ][ ] { }{ } [ ] [ ] [ ][ ]{ }n

gnmmmm

mmnmnnngnmm

Tmmnmnnnm

BUSVBAX

=[ ] { } { } { }[ ]

{ }{ }{ }

321 vvv

[ 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

Matrix Notation

A matrix [A] can be described using row,column as

54535251

44434241

34333231

24232221

14131211

aaaaaaaaaaaaaaaaaaaa

( row , column )

[A]T -Transpose - interchange rows & columns[A]H - Hermitian - conjugate transpose

Matrix Notation

Square

5554535251

4544434241

3534333231

2524232221

1514131211

aaaaaaaaaaaaaaaaaaaaaaaaa

5545352515

4544342414

3534332313

2524232212

1514131211

353433

25242322

1514131211

a0000aa000aaa00aaaa0aaaaa

Triangular Diagonal

Symmetric Vandermonde Toeplitz

aa1aa1aa1aa1

A matrix [A] can have some special forms

Linear Algebra Rules & Definitions

General Matrix

nm1n1n

m12221

m11211

aaaaaa

L Column Vector

Row Vector

mj21 ccccC LL=

AdditionScalar MultiplierMatrix Multiplication

ijijij bac]B[]A[]C[ +=⇒+=

ijij a*sb]A[*s]B[ =⇒=

{ }jiij bac]B][A[]C[ =⇒=

∑=⇒

kjikij

ik2i1iij bac

Matrix Manipulation

A matrix [C] can be computed from [A] & [B] as

3534333231

2524232221

1514131211

cccccc

bbbbbbbbbb

aaaaaaaaaaaaaaa

5125412431232122112121 bababababac ++++=

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

aaadaaad

Post-Multiplication by a Diagonal Matrix

dD]A[MMM

Transpose of a Matrix

]A[]B[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

=+−=−+−

−−−

110121

This set of equations has a unique solution

whereas this set of equations does not

2y2x42z1y2x

=−=−+−

−−−

024121

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

xxxxx0xxxx00xxx000xx0000x

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;

l;ulau ∑∑∑ −=−

The simple 3x3 stiffness matrix can be decomposed to be

−−=

−−−

0.115.1

1667.00.015.0

Each of the factors are given by

[ ] [ ][ ]

−−−−

∑∑∑∑

133133221331321131

1331231221221121

131211

aaaaaaaaaaaaaaaaa

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

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]

[ ] { } { } { }[ ]

{ }{ }{ }

321 vvv

[ ] { } { } { } { } { } { } L+++= T333

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

u 22 =

Then the matrix [A2] can be formed to be

[ ] { } { } [ ]{ }

−−−−

111111111

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

The SVD of matrix [A3] is

[ ]{ }{ }{ }

000111321

These are the independent quantities that make up the matrix which has a rank of 2

[ ] { } { } { }TTT 0000000

3211321

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

ν i ν j

{ } [ ]{ }

γγγεεε

τττσσσ

⇒ε=σ

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

−−−−

−−

L4L6L2L6L612L612

L2L6L4L6L612L612

−−

−−−−

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\

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

SVD of the input excitation matrix identifies the rank of the matrix - that is an indication of how many linearly independent inputs exist

[ ] { } { } { }[ ]

{ }{ }{ }

21xx 0vv

SVD of Multiple Reference FRF Data

SVD of the [H] matrix gives an indication of how many modes exist in the data

[ ] { } { } { }[ ]

{ }{ }{ }

321 vvv

FREQUENCY RESPONSE FUNCTIONS

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

( )[ ] [ ]( )

[ ]( )*

−= ∑

Extended analysis and evaluation of systems

[ ][ ] [ ][ ][ ]2I `UMUK ω=

[ ][ ][ ] [ ] [ ][ ][ ]2I

T `UMUUKU ω=

[ ] [ ] [ ] [ ][ ][ ][ ][ ] [ ][ ] [ ][ ][ ] [ ][ ]TI

MUUKMUUK

VK`VKK

−−

[ ] [ ] [ ] [ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ]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 -...

Documents

teachers and studentsof algebra 1, algebra 1 honors and...

a texas solution to openstax: a national problem · algebra...

prentice hall: pre-algebra © 2001, algebra i, algebra ii

algebra 3: algorithms in algebra

solid mechanics review 061904 - faculty server...

algebra super algebra superior_2.pdfior 2

algebra exterior y algebra de grassmann

foundational pre-algebra pre-algebra transitional algebra...

algebra basics algebra basics algebra basics algebra basics

fach / materia: arithmetik & algebra / aritmetica & algebra

analytical model improvement localization...

pre-algebra * algebra and geometry basics * algebra i · pdf...

fİzİksel aktİvİtenİn tevİkİnde · 2015. 1. 5. ·...

algebra rules! algebra

“algebra“ > · vollbild > < > beenden...

honors algebra 2 algebra 1 review - hasd.us 1 review...

algebra i 1nociones-preliminares-algebra

algebra expressions really, what is algebra?

system modeling concepts 071206 -...

mecatronica.espe.edu.eclgebra-lineal.pdf · algebra lineal...