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can anybody tell me the condition number of $X \cdot Y$ with $X \in \mathbb{R}^{M \times N}, Y \in \mathbb{R}^{N \times P}$.
I tried to describe the matrix multiplication by a sum of many multiplications ($a=M \cdot N \cdot P$), which each has a condition number of $\kappa=\frac{(x_i+y_i)^2}{|x_i y_i|}$, but do not know how to connect them.

Or may it be just the product of the condition numbers of the two matrices: $\kappa_1 \cdot \kappa_2 $ ?

Bamahama
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  • What definition of condition number do you have for non square (singular) matrices? – flawr Sep 10 '15 at 09:36
  • Isn't there a condition number for non square matrices? I think about the relative condition which tells me something about maximal error the matrix matrix product can have. – Bamahama Sep 10 '15 at 13:25
  • Do you want the condition of $(X,Y) \mapsto XY$ or the condition of $X\mapsto XY$ for a fixed $Y$? – user251257 Sep 10 '15 at 14:49
  • The first one, so you were right with your answer – Bamahama Sep 10 '15 at 23:13

1 Answers1

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I will write it using the Frobenius norm $\|\cdot\| = \|\cdot\|_F$, as it makes the result looks more like the scalar result. Also, I use the norm $\|(X,Y)\| = \sqrt{ \|X\|^2 + \|Y\|^2 }$ on $\mathbb R^{M\times N} \times \mathbb R^{N\times P}$.

Let $f(X,Y) = XY$. Then, the differential of $f$ is given by $$ Df(X,Y)[H,K] = XH + KY $$ Thus, for $\|(H,K)\| = 1$ we have \begin{align} \| Df(X,Y)[H,K] \| &\le \|XH\| + \|KY\| \\ &\le \|X\|\|H\| + \|K\|\|Y\| \\ &\le \sqrt{ \|X\|^2 + \|Y\|^2 } \sqrt{ \|H\|^2 + \|K\|^2 } = \|(X,Y)\|. \end{align} That is, the operator norm of $Df(X,Y)$ is bounded by $\|(X,Y)\|$.

Thus, the condition number is bounded as following: $$ \kappa_f(X,Y) = \frac{\|Df(X,Y)\|\|(X,Y)\|}{\|f(X,Y)\|} \le \frac{\|(X,Y)\|^2}{\|XY\|} = \frac{\|X\|^2 + \|Y\|^2}{\|XY\|}. $$

user251257
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