Relation between perturbed matrix and condition number of the matrix

Question

If A is non‐singular but the perturbed matrix (A+δA) is singular, then show that  $$∥A∥/∥δA∥≤y $$ Where y is condition number of the matrix A.

Tried for a solution

The relation $$(A+δA)(x+δx)=b $$ implies that $$Ax+δA(x+δx)+Aδx=b$$

and since $$Ax=b$$

the above becomes $$δA(x+δx)+Aδx=0$$

and hence $$δx=−A^{−1}δA(x+δx)$$

which implies that $$∥δx∥≤∥A^{−1}∥∥δA∥∥x+δx∥=y⋅\frac {∥δA∥}{∥A∥} ∥x+δx∥,$$

and finally

$$\frac {\frac{∥δx∥}{∥x+δx∥}} {\frac {∥δA∥}{∥A∥}}≤y$$

$$\frac{∥δx∥}{∥x+δx∥} {\frac {∥A∥}{∥δA∥}}≤y$$

This what I get so far ! I don't know how to remove $$\frac{∥δx∥}{∥x+δx∥}$$

Answer 1

If $A+E$ is singular, $(A+E)x=0$ for some $x\neq 0$, so $$ x=-A^{-1}Ex\implies\|x\|\leq\|A^{-1}\|\|E\|\|x\|\implies 1\leq\|A^{-1}\|\|E\| \implies\frac{\|A\|}{\|E\|}\leq\|A^{-1}\|\|A\|. $$

Answer 2

You cannot conclude because you do not kwow what do you want to prove.

1. Line 16, you give an upper bound of $\delta x$. In fact this bound can be reached (think about eigenvectors) and then, is the max-error (denoted by $\Delta x$) with respect to $x$.

2. Thus $||\Delta x∥\approx y\dfrac {∥δA∥}{∥A∥} ∥x+\Delta x∥$ implies $\dfrac{||\Delta x||}{||x||}\approx y\dfrac {∥δA∥}{∥A∥}$ because $x+\Delta x\approx x$.

3. Finally, if $y=10^k$ anf if $A$ is known with $l$ significant digits, then $\dfrac{||\Delta x||}{||x||}\approx 10^{k-l}$ and $x$ is known with $l-k$ significant digits (we lose $k$ digits). Note that (for example in the case of Hilbert matrices) one may have $k>l$.