Condition number of a rectangular matrix

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?

1 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 .

• So the generalized definition is generally accepted? And my original understanding is only correct under the use of the euclidean norm? Are other norms than the euclidean widely used? Apr 28, 2013 at 19:44
• Ya others norms can be used. In a finite dimensional space all norms are equivalent. For more details please go through this link en.wikipedia.org/wiki/Matrix_norm Apr 28, 2013 at 19:50