# Condition number of $X^{T}AX$

$A$ 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?

Let's show that we cannot bound the condition number of $X^T A X$ by using only the condition number of $A$ and the norm of $X$.
Let $A=I$, so its condition number is exactly $1$.
Let $X$ consist of an invertible diagonal block with $p$ rows of zeros padded at the bottom:
$$X = \begin{pmatrix} D \\ 0 \end{pmatrix}$$
Now $X^T A X = D^2$, and its condition number would be $\|D^{-2}\|\cdot \|D^2\|$ for some choice of matrix norm. While the norm of $X$ can be used to bound the norm of $D^2$, we have no similar control over how large the norm of $D^{-2}$ can be.
• Thanks. I have access to more information about the properties of $X$ which restricts $||D^{-2}||$. I am more interested in how the condition number of $X^{T}AX$ can be bounded using condition number of $A$ and the condition number defined based on the pseudoinverse of $X$. – gpavanb Feb 5 '16 at 19:43