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: