Proving the Inequality for Pseudoinverse and 2-Norm: Is ||A+|| ≤ ||A1-1||?

[Codezion]

Homework Statement

Prove that the ||A⁺|| \leq ||A₁^-1||
Where A⁺=(A^TA)^-1A^T, ||.|| is the 2 norm and A is an mxn matrix

Homework Equations

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

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:
||(A1^TA1 + A2^TA2)^_1(A1 A2)|| but cannot proceed much. I really appreciate any help! Thank you.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[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?

 

[Codezion]

Thanks for the reply!
what is A dagger?

Here is the actual question: http://books.google.com/books?id=bj-Lu6zjWbEC&printsec=frontcover#PPA85,M1 (look at 11.1)

Thank you!

 

[D H]

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.

 

[Codezion]

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