How can the pseudoinverse of a rectangular matrix use SVD like an inverse?

[Cyrus]

Hi, I need help with the pseudoinverse. I would like to know what it is and what it does. From what I've found on websites, you can find the inverse of a nonsquare matrix. My book, "Linear Algebra and its applications" by David Lay, says that the reduced singualr value decomposition can be written as:

$$A = U_r (D)V_r^T$$

and that the pseudo inverse can be written as:

$$A^+ = V_r (D^-^1) U_r^T$$.

It looks as if the author simply did the inverse of A to get the pseudoinevrese. But how could he do that when A is rectangular?

Thank you,


Cyrus

 

[Cyrus]

please help!

 

[HallsofIvy]

For non-square A, you can set up the decomposition so that D is square:
If A is 2 by 3, then U would have to be 2 by 3 but D and V would be 2 by 2.

You take the inverse of D but you only need the transposes of U and V.

Since the transpose of U would be 3 by 2, the pseudo-inverse of A, A⁺, would be 3 by 2.

 

[Cyrus]

why do you take the transpose of U and V. They are no longer orthogonal matricies because we cut out some of them when we partitioned it. So it is no longer true that their inverse equals their transpose.

 

[Cyrus]

anyone can anwser my question please?