GATE-2018 (24) :Wyllie – Time average equation

GATE-2016

Q51. The singular value Decomposition of a square matrix ‘J’ is given by J= UλV^T. The inverse of matrix ‘J’ will be

      A)     J^-1 = Uλ^-1 V ^T

      B)  J^-1 = Vλ^-1U^T

      C)     J^-1 = U^Tλ^-1V^T

      D)   J^-1 = Uλ^-1V

Solution:

A computing the  inverse of a matrix using SVD

A square matrix A is non singular if σ_i not equal to zero , for all i

A is a n*n nonsingular matrix, then its inverse is given by

A= UDV^T

A^-1= VD^-1U^T

       

Where D^-1 =diag (1/σ1, 1/σ2………………1/σn)

If A is singular or ill-conditioned, then we can use SVD to approximate its inverse by the following matrix

A^-1 = (UDV^T) ^-1

     = VD^-1U^T

   D^-1= { 1/σi , if σi >t

            O , otherwise

( where t is a small therehold)

J= UλV^T

The inverse of matrix J= Vλ^-1U^T

J^-1=VλU^T