Computing the condition number of a matrix

Question

Given$$A=\begin{bmatrix}23.89&-36.48&1.432&21.65\\-36.48&54.58&-5.193&-34.45\\1.432&-5.193&-1.0717&1.937\\21.65&-34.45&1.937&20.50\end{bmatrix}.$$ I am given the eigenvalues of $A$ as $100$, $1$, $-0.1$, $-3$, which matches with an error of $10^{-3}$ the eigenvalues given by octave

A = [23.89 -36.48  1.432   21.65; 
    -36.48  54.58 -5.193  -34.45;
     1.432 -5.193 -1.0717  1.937;
     21.65 -34.45  1.937   20.50] ;

>> lambdas=eig(A)
lambdas =

   -3.001013
   -0.098193
    1.001209
   99.996297

With the definition of the conditional number $\kappa$: $$\kappa_{2}(A) = \frac{|\lambda_{n}|}{|\lambda_{1}|},$$ I can compute the conditional number with:

kappa = abs(max(lambdas)) / abs(min(lambdas))

which gives the answer $33.333$.

However when I check the answer with the following octave command it does not match:

cond(A)

which gives $1018.4$. What am I doing wrong?

Answer 1

The following is wrong. Corrected below

---

The condition number (in $L_2$ norm ) is the ratio of the maximum/minimum singular values.

This equals the ratio of the maximum/minimum (absolute values of) eigenvalues only if the matrix is symmetric (or more general, normal) which is not the case here.

>> svd(A)
ans =

   99.996297
    3.001013
    1.001209
    0.098193

>> 99.996/0.098
ans =  1020.4

---

This matrix is normal , hence, effectively, the ratio of maximum/minimum absolute value of eigenvalues gives the number condition. But you must first take the absolute value, then select the extremes.

> kappa = max(abs(lambdas)) / min(abs(lambdas))
kappa =  1018.4

This was rightly pointed out by Parcly Taxel's comment.