Constructing Singular ValueDdecomposition for a Product of two Matrices

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Given real matrices A,B with appropriate dimensions to compute the product AB
and given the SVD of matrix A:
$$ A=U_A S_A V_A^T $$



Is there a way or method to construct SVD(AB) analytically?
$$ A=U_AB S_AB V_AB^T $$



I am mainly interested in the relation between orthonormal bases V_A and V_AB.
There has to be a transformation (rotation?) T that depends on matrix B (and maybe also A).
$$ V_AB = T V_A $$
How can I compute T ?



In my specific case B is a lower-triangular matrix obtained by a Choleski Decomposition. Eventually this simplifies the problem.



I already checked a number of old papers from Golub, Paige, Stoer, Moor, ... but that didn't really help me.



https://epubs.siam.org/doi/pdf/10.1137/S0895479897325578



https://epubs.siam.org/doi/pdf/10.1137/0613060



Thanks for your support!




matrix-decomposition orthonormal svd




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    up vote
    -1
    down vote

    favorite












    Given real matrices A,B with appropriate dimensions to compute the product AB
    and given the SVD of matrix A:
    $$ A=U_A S_A V_A^T $$



    Is there a way or method to construct SVD(AB) analytically?
    $$ A=U_AB S_AB V_AB^T $$



    I am mainly interested in the relation between orthonormal bases V_A and V_AB.
    There has to be a transformation (rotation?) T that depends on matrix B (and maybe also A).
    $$ V_AB = T V_A $$
    How can I compute T ?



    In my specific case B is a lower-triangular matrix obtained by a Choleski Decomposition. Eventually this simplifies the problem.



    I already checked a number of old papers from Golub, Paige, Stoer, Moor, ... but that didn't really help me.



    https://epubs.siam.org/doi/pdf/10.1137/S0895479897325578



    https://epubs.siam.org/doi/pdf/10.1137/0613060



    Thanks for your support!




    matrix-decomposition orthonormal svd




    share|cite|improve this question





















      up vote
      -1
      down vote

      favorite









      up vote
      -1
      down vote

      favorite











      Given real matrices A,B with appropriate dimensions to compute the product AB
      and given the SVD of matrix A:
      $$ A=U_A S_A V_A^T $$



      Is there a way or method to construct SVD(AB) analytically?
      $$ A=U_AB S_AB V_AB^T $$



      I am mainly interested in the relation between orthonormal bases V_A and V_AB.
      There has to be a transformation (rotation?) T that depends on matrix B (and maybe also A).
      $$ V_AB = T V_A $$
      How can I compute T ?



      In my specific case B is a lower-triangular matrix obtained by a Choleski Decomposition. Eventually this simplifies the problem.



      I already checked a number of old papers from Golub, Paige, Stoer, Moor, ... but that didn't really help me.



      https://epubs.siam.org/doi/pdf/10.1137/S0895479897325578



      https://epubs.siam.org/doi/pdf/10.1137/0613060



      Thanks for your support!




      matrix-decomposition orthonormal svd




      share|cite|improve this question











      Given real matrices A,B with appropriate dimensions to compute the product AB
      and given the SVD of matrix A:
      $$ A=U_A S_A V_A^T $$



      Is there a way or method to construct SVD(AB) analytically?
      $$ A=U_AB S_AB V_AB^T $$



      I am mainly interested in the relation between orthonormal bases V_A and V_AB.
      There has to be a transformation (rotation?) T that depends on matrix B (and maybe also A).
      $$ V_AB = T V_A $$
      How can I compute T ?



      In my specific case B is a lower-triangular matrix obtained by a Choleski Decomposition. Eventually this simplifies the problem.



      I already checked a number of old papers from Golub, Paige, Stoer, Moor, ... but that didn't really help me.



      https://epubs.siam.org/doi/pdf/10.1137/S0895479897325578



      https://epubs.siam.org/doi/pdf/10.1137/0613060



      Thanks for your support!





      matrix-decomposition orthonormal svd





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