[Math] How to find lower bound on condition number of a matrix

linear algebraMATLAB

The condition number is given as follows:
$c(B)=\|B\|\|B^{-1}\|$

My questions are:
1. Considering that $B$ is non singular, i have to show a lower bound on $c(B)$ for $B$
2. What is that lower bound if $B=I$?
3. I have this matrix $$C=\begin{bmatrix}
1 &0.999 \\
1 &1.001 \\
\end{bmatrix}$$
need to evaluate its condition number using Matlab. (I know that we can use "cond" command for this part, just do not know how")
4. Need to find a vector v around which the solution to $Cv = b$ is not stable

Thanks

Best Answer

Hints:

  1. Note that if $\|\cdot\|$ is a matrix norm, then $\|\textrm{some matrix product}\|\le\|B\|\|B^{-1}\|$.
  2. $\|I\|=\|I\,I^{-1}\|$, so ...
  3. help cond
  4. Consider $v=(1,1)^T$. What is $b$? If we make a small change in $b$, so that $b$ becomes $(2,2)^T$, how would $v$ change? If we don't change $b$, but change the coefficients of $C$ slightly instead, so that $C$ becomes $\begin{pmatrix}1&1\\ 1&1\end{pmatrix}$, what happens then?
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