From what I understand, the condition number of a rectangular matrix $A$ is its largest singular value divided by its smallest nonzero singular value
$$\kappa(A) := \frac{\sigma_1 (A)}{\sigma_n (A)}$$
Where $\sigma_1 (A)$ is the operator norm of $A$ and $\sigma_n (A)$ is the operator norm of $A^\dagger$, the pseudoinverse of A. Is this correct and is it generally accepted?
I have not been able to find much on the subject online. Is this used in any notable applications?
Best Answer
Condition number of square nonsingular matrix $A$ is defined by
Definition $1$: $cond(A) = \|A\| \|A^{-1}\|$, where the norm $\|.\|$ above could be any of the norms defined for matrices. If we use the usual Euclidean norm on vectors and the associated matrix norm, then the condition number is the ratio of the largest singular value of matrix A to the smallest.
Definition $2$: Condition number for any matrix is defined as:
$cond(A) = \|A\| \|A^+\|$, where $A^+$ is the pseudo inverse of the matrix $A$. Note that for a square non singular matrix $A^+ = A^{-1}$ which implies that Definition $2$ is the generalization of Definition $1$ to find out condition number of any matrix .