$A$ is a symmetric matrix and is known to be invertible. $X$ is rectangular of size $(N+p) \times N$ with $p > 0$ but full column rank. Can we provide an upper bound on the condition number of $X^{T}AX$ based on this information along with the condition number of $A$ and norm of $X$?Can we use the pseudoinverse of $X$? Do submultiplicative norms directly extend to them?
Condition number of $X^{T}AX$
linear algebra;condition number