# Relation of matrix / operator norm to rank deficiency?

(This is a follow-up to Criterion for detecting rank-deficiency via QR decomposition?) To recap:

I'm solving a system like $P \approx X Y^T$, where P is a large sparse matrix, and X and Y are a low-rank (rank $k$) factorization of P. I need to solve, for example, for a row of X, $X_u$, given $P_u$ and $Y$. I'm approximating with $X_u = P_u Y (Y^T Y)^{-1}$.

I'm computing a solution for $(Y^T Y)^{-1}$ with a QR decomposition. But the matrix may be rank deficient or very nearly so. If it's nearly so, the solution for $X_u$ can be, for example, quite large -- a "bad" answer in the sense that it's much larger than any other row of $X$.

To take this in a different direction: it's generally speaking true in my problem that the matrix $A = (Y^T Y)^{-1}$ ought to send vectors that are about length 1 to vectors that are about length 1. In the normal general case that's what one observes and is expected. Except it doesn't happen when the input is bad: too small for the given rank, etc.

I understand the idea of a standard operator norm on matrices -- the most that the matrix as a linear operator will "lengthen" its input -- the smallest $c$ such that $\|Av\| \le c\|v\|$ for all non-degenerate $v$. This seems relevant.

As I understand, this $c$ is just the largest singular value of $A$, which is easy to find. I can use this as a criterion for detecting "bad" input.

However I'm missing a great deal of rigor here. Does this idea have any legitimate bearing on the problem I'm trying to solve? Am I missing basic, related results?

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