Pseduoinverse and 2-norm

1. Feb 2, 2009

Codezion

1. The problem statement, all variables and given/known data
Prove that the ||A+|| $$\leq$$ ||A1-1||
Where A+=(ATA)-1AT, ||.|| is the 2 norm and A is an mxn matrix

2. Relevant equations

A = [$$\stackrel{A1}{A2}$$] where A1 is an nxn nonsingular square matrix and A2 is any random matrix that is (m-n)xn

3. The attempt at a solution

All I did was replace all the A's in the pseduoinverse with A1 and A2 and found the following:
||(A1TA1 + A2TA2)_1(A1 A2)|| but cannot proceed much. I really appreciate any help! Thank you.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

Last edited: Feb 3, 2009
2. Feb 2, 2009

MathematicalPhysicist

Well, (A^TA)^-1=A^-1(A^T)^-1
So you get that A dagger equals A^-1 by definition, have you wrriten the problem as it is in the book?

3. Feb 2, 2009

4. Feb 2, 2009

D H

Staff Emeritus
You did not write the problem as specified in the book.

The book asks you to show that $$||A^+||_2 \le ||A_1^{-1}||_2$$ . This is not the same as $$||A^+||_2 \le ||A^{-1}||_2$$ . The latter does not even make sense because A can not have an inverse.

5. Feb 3, 2009

Codezion

Thank you DH! I have corrected my problem - I think all the latex syntax confused me.