Spectral norm of psuedoinverse of a matrix

Question

I have a symmetric $d\times d$ matrix, such that all entries are either +1 or -1, therefore the diagonal entries are +1. I want to upper bound the spectral norm of the psuedoinverse of such a matrix. I did some simulations on matlab, and I find that the spectral norm of psuedoinverse is always less than or equal to 1. If this is indeed true, how do I show it, if not, what is a counterexample?

Answer 1

Random counterexample: $$ A=\pmatrix{ 1& 1&-1& 1& 1\\ 1& 1& 1&-1& 1\\ -1& 1& 1& 1& 1\\ 1&-1& 1& 1& 1\\ 1& 1& 1& 1& 1}. $$ The five eigenvalues of $A$ are $-2,\,\frac{3-\sqrt{17}}{2},\,2,\,2$ and $\frac{3+\sqrt{17}}{2}$. Since $|\lambda|_\min(A)=\left|\frac{3-\sqrt{17}}{2}\right|\approx0.56<1$, we have $\|A^{-1}\|_2=\frac{1}{|\lambda|_\min(A)}>1$.