I feel like I'm so close to solving this... I think I'm missing a step!?
I am to show that X = the pseudoinverse of A (I'll denote A+) will minimize ||AX - I||_F and find the value of the minimum...
I started with ||AA+ - I||_F = (Tr((AA+ - I)(AA+ - I)*))^1/2 (by definition)
After some pseudoinverse properties and expansion, it boils down to
=(Tr(I - AA+))^1/2
Then I looked at the SVD of A, A = USV* and A+ = VE+U*
=(Tr(I - USV*VS+U*))^1/2 = (Tr(I - USS+U*))^1/2 because V is orthogonal... and here is where I'm stuck... please help!