     # Diagonalization of Matrices - .Diagonalization of Matrices •Motivation •Eigenvalue decomposition

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• Diagonalization of Matrices

By: Abdurazak Mudesir

• Diagonalization of Matrices

Motivation Eigenvalue decomposition Singular-value decomposition Discrete Fourier transform Applications : FEXT cancellation

Wireless applications

• Motivation

Image Compression

Noise Filtering

And more..

• Eigenvalue decomposition

Definition: Let A be any square matrix. A scalar is called an Eigenvalue of A if there exists a nonzero (column) vector v such that Av =v

Any vector satisfying this relation is called an eigenvector of A belonging to the eigenvalue

• EVD cont.

An n n matrix A is similar to D=diag(d1,d2,..dn) iff A has n linearly independent eigenvectors.

The (d1,d2..dn ) are the corresponding eigenvalues

P is the matrix whose columns are the eigenvectors

**

APPD 1=

1= PDPA

• Algorithm to solve EVD

1) Find the eigenvalues of the matrix2) Find n linearly independent eigenvectors.3) Construct P from the vectors in step 24) Construct D from the eigenvalues

• Example:

• Singular-value decomposition

Any m by n matrix A may be factored such that A = UVT

U: m by m, orthogonal, columns are the eigenvectors of V: n by n, orthogonal, columns are the eigenvectors of : m by n, diagonal, the singular values are the square roots of the

eigenvalues of both and

SVD of Aunitarily (orthogonally) equivalent to the

diagonal matrix .

TAAAAT

TAA AATTVUA =

• proof.

andTVUA =

2=VAVAT

TT UVA =TTT VUUVAA =

TT VVAA 2=2= VAVAT

• Algorithm to find SVD

1) Find the eigenvalues of the matrix ATA and arrange them in descending order

2) Find the number of nonzero eigenvalues of the matrix

1) Find the Orthogonal vectors of corresponding to the eigenvalues above ( arrange the same order to form the V matrix.)

2) Form the diag matrix .3) Find the first column-vectors of the matrix U(mxm)

AAT

AAT

• Algorithm cont.

6) Add to the matrix U the rest of vectors (they must be orthogonal to the r vectors)

use the Gram_Schmidt Orthogonalization process.

rm

• Examples

• DFT

)mod][]([]y[1

0

NknhkxhxnN

k

=

==

)mod][(],[: NknhnkTLet =Txy =

• Toeplitz matrix

=

..)2()1(...

........

.)0()1()2()1()0()1(

...)2()1()0(

NhNh

hhhnhhhNhNhh

T

• DFT Matrix

WW*=I unitary

=

)1)(1()1(21

642

321

.....1...1...1...1

...1

...1

.....1111

1

NNN

NN

NN

NNN

NNN

WWW

WWWWWW

NW

• Y=XH ,(DFT domain)

Wy=WTx=DX=DWx WTx=DWx WT=DW

=

)1(0...

........

.....00.0)2(0.00)1(0

.....000)0(

NHo

HH

H

D

DWTW =1

• Conclusion: A DFT matrix deagonalizes a toeplitz matrix

DWWT 1=

• Applications:

FEXT cancellation.The Next slides are taken from prof Henkels

presentation.

• MIMO systemsA possibility to increase data rates without

boosting the power

1a2a

1MaMa

1c2c

1McMc

1b2b

1LbLb

...

CabinetCentral office

FEXT

NEXT

ka kc.........

cablebundle

I

cablebundle

II

11,kj

FEXT

K Lmjk kk k

mj jk

Nk

m

EXT

c bha a gh==

= + + 14442444314444244443

• MIMO-SystemeFunk Smart Antennas

1a2a

Ma

1c2c

Mc

( )1

tkjK

jjk

h ac=

=

• MIMO systemsSingular-value decomposition

MIMO equationsfor each frequency: ( ) ( ) ( ), 1,...,n n n n N= =y A x

system matrices ( )no diagonalare for aFEX ll , .,T 1..n n N= A

Singular-value decompositions (SVDs):

( ) ( ) ( ), 1,( ) ...,Hn n n nn N= =A Q P

( ), ( ) ... unitary matrices ... ( ) diagonalreal matrices with elements 0n

n n K KK K

P Q

• MIMO-SystemeSingular-value decomposition

( ) ( ) ( ), 1,( ) ...,Hn n n nn N= = P

Inverse: ( )nP

Inverse: ( )H nQ

A Q

( ), ( ) ... unitary matrices ... ( ) diagonalreal matrices with elements 0n

n n K KK K

P Q

• MIMO systemsBlock diagram of the procedure

T-Part 1

T-Part KR-Part K

R-Part 1

MIMO

channel

-

K loops

(1)P(2)P

( )NP

(1)HQ

(1)t

( )Nt

(1)r

( )Nr (2)HQ

( )H NQ

( ) ( ), ( 1,...,)nn n n N= =r t

• MIMO systemsGain in capacity

0 500 1000 1500 20000

20

40

60

80

100

120

140

160

Distance CO to cabinet [m]

Bitr

ate

per l

oop

[Mbi

t]

MIMO

Non-MIMO

• Reference: http://www.coastal.edu/~jbernick/ http://www.cs.ut.ee/~toomas_l/linalg/lin2/node14.html http://web.mit.edu/be.400/www/SVD/Singular_Value_Decomposition.htm http://www.cs.utk.edu/~dongarra/etemplates/node43.html

http://www.cs.utk.edu/~dongarra/etemplates/node43.html

Diagonalization of MatricesDiagonalization of MatricesMotivationEigenvalue decompositionEVD cont.Algorithm to solve EVDExample:Singular-value decompositionproof.Algorithm to find SVDAlgorithm cont.ExamplesDFTToeplitz matrixDFT MatrixConclusion: A DFT matrix deagonalizes a toeplitz matrixApplications:MIMO systemsA possibility to increase data rates without boosting the powerMIMO-SystemeFunk Smart AntennasMIMO systemsSingular-value decompositionMIMO-SystemeSingular-value decompositionMIMO systemsBlock diagram of the procedureMIMO systemsGain in capacityReference:

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