3
$\begingroup$

$A$ 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?

$\endgroup$
6
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 $X$. $\endgroup$ – gpavanb Feb 5 '16 at 19:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.