relation between size of matrix and condition number

Question

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) for A. I understand that the pseudo inverse of A is only possible when rank(A) is greater than equal to M. But, I want to know if there is anyway to reason about the condition number cond(A) with that of NxM. In other words, how is the condition number related to the rank of the matrix?

Answer 1

I understand that the pseudo inverse of A is only possible when rank(A) is greater than equal to M.

This is incorrect. The Moore-Penrose pseudoinverse is defined for all matrices, and Matlab's pinv implements it for all matrices. The pinv command description has an example of this sort. It's true that the computation may be less expensive for matrices of full rank, but this is a different issue (and in any case, it seems that Matlab goes through SVD regardless of the rank).

Is there any relationship between size of a matrix A with the condition number ?

No. The identity matrix has condition number $1$, regardless of its size. On the other hand, $$A=\begin{pmatrix} 1 & 0 \\ 0 & 10^{-9}\end{pmatrix}$$ is a $2 \times 2$ matrix of rank $2$, yet it has cond(A) of about $10^9$.