Matrix inverse with svd

Hello,

I am trying to invert a matrix using svd and \ and comparing the result with inv().

For this matrix:

A = [ 2 4 6; 8 5 7 ; 0 3 1]

I did the svd decomposition as follows:

[U,S,V] = svd(A);

And, I believe the inverse of A can be expressed as :

So, I tried it by writing the code below:

disp('inverse with svd:')
disp(V*S\U')

And, I got this answer:

inverse with svd:
    0.0236    0.0440   -0.0528
    0.1765    0.1361    0.1922
    0.4277   -0.3787   -0.1247

Then, I tried to invert A using inv() as follows:

disp('inverse with inv:')
disp(inv(A))

However, I got a different answer as seen below:

inverse with inv:
   -0.2000    0.1750   -0.0250
   -0.1000    0.0250    0.4250
    0.3000   -0.0750   -0.2750

Can someone tell me what I am doing wrong?

My main goal is that I want to use svd to invert large matrices that may be close to becoming singular.

Thank you

Answer 1

[Christine Tobler]

Use one of

A = [ 2 4 6; 8 5 7 ; 0 3 1]

A = 3×3

2 4 6 8 5 7 0 3 1

[U,S,V] = svd(A);
V*inv(S)*U'

ans = 3×3

-0.2000 0.1750 -0.0250 -0.1000 0.0250 0.4250 0.3000 -0.0750 -0.2750

V*(S\U')

ans = 3×3

-0.2000 0.1750 -0.0250 -0.1000 0.0250 0.4250 0.3000 -0.0750 -0.2750

% The formula you're using above puts the parentheses in a different place
% than you think:
V*S \ U'

ans = 3×3

0.0236 0.0440 -0.0528 0.1765 0.1361 0.1922 0.4277 -0.3787 -0.1247

(V*S) \ U'

ans = 3×3

0.0236 0.0440 -0.0528 0.1765 0.1361 0.1922 0.4277 -0.3787 -0.1247

[Elie Hatem]

Thank you so much!

[Arnab Paul]

What happens to non-square matrix?

is it same as pinv(A)?