Calculate effective rank of matrix How can I calculate the effective rank of a matrix? I know how to calculate the rank but not how to calculate the effective rank.
 A: I've never heard of the effective rank of a matrix. However, you might probably mean the numerical rank. For a matrix $A$ with singular values $\sigma_1\geq\sigma_2\geq\cdots\geq \sigma_n\geq 0$, the $\epsilon$-numerical rank could be defined as
$$
r_{\epsilon}=\min\{r:\;\sigma_r\leq\epsilon\}.
$$
So, the relatively small singular values are considered to be zero depending on the given "tolerance" $\epsilon$. Usually (when deciding whether or not a given matrix is numerically rank-deficient or not), $\epsilon\approx u\sigma_1$, where $u$ is the machine precision.
A: You might be interested in the following publication:
Olivier Roy and Martin Vetterli, The effective rank: A measure of effective dimensionality, 15th European Signal Processing Conference, 2007, available at https://infoscience.epfl.ch/record/110188/files/RoyV07.pdf.
They define "effective rank" as the entropy of the notional distribution obtained by normalising the singular values. The $\ell^1$ norm of the singular values is called the nuclear norm.
It has the property that for an m x n matrix A,
1 <= erank(A) <= rank(A) <= min(m,n)

It has other pleasant properties, and a (reasonably) intuitive geometric interpretation in terms of linear transformations.
A: I saw the use of effective rank of a matrix in the paper Vishwanathan, S. Vichy N., et al. "Graph kernels." The Journal of Machine Learning Research 11 (2010): 1201-1242.
There it is mentioned that the effective rank of a matrix is the number of distinct eigenvalues.
Eigenvalues of a matrix $A$ can be obtained by solving $det(A-\lambda I) = 0$
