$A$ 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?


1 Answer 1


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.

  • $\begingroup$ 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, 2016 at 19:43

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