Computing the condition number of a matrix
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 99.996297With 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?
$\endgroup$ 31 Answer
$\begingroup$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 = 1020.4This 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 = 1018.4This was rightly pointed out by Parcly Taxel's comment.
$\endgroup$
