Pseudoinverse of matrix...

[vkekk]

what is pseudoinverse matrix and in what way it differ from inverse matrix.

For example , The pseudoinverse of channel matrix, H is

H' = ( H*H)^-1 H*

Where H' ---- pseudoinverse of channel matrix, H.

Can anybody explain me to understand the above equation (i.e in what way pseudoinverse work , ( H*H)^-1 H* )

 

[abaidullah]

Please refer this page i think it would help you.

https://en.wikipedia.org/wiki/Moore-Penrose_pseudoinverse

Added after 3 minutes:

have a look at this page also.

**broken link removed**

 

[eroica]

Let y = Hx be your system equation where x is the input and y is the output.
If you have inv(H), then your system equation is easily solved by doing x = inv(H)y. However, we cannot have inv(H) directly in many of the real applications. That is where the pseudo inverse comes from.

Although you don't have inv(H), you can change y = Hx to H'y = H'Hx which in turn x = ((H'H)^-1)Hy. In deed, x is the famous 'least square approximation'. Therefore, to be precise, x_ls = ((H'H)^-1)Hy where x_ls is the least square solution and ((H'H)^-1)H is the pseudo inverse of H.