Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-diagonal elements change from 0 to 1, the information contained in the matrix reduces continuously.
Example: considering a matrix $A(a)$ \begin{pmatrix} 1&0&0&\\ 0&1&a\\ 0&a&1 \end{pmatrix}. How do I define an information measure $I(A(a))$ that is continuous to $a$ for matrix $A(a)$, so that $I(A(0)) = I(I_3) = 3$ and $I(A(1)) = I(I_2)= 2$?
Rank is not continuous; Shannon information entropy on eigenvalues does not give desired values; and von Neumann entropy $S(\rho) = -tr(\rho\log(\rho))$ equals zero for any identity matrix while the dimension information lost; and $S(A(1))$ from my example does not reduce to S($I_2$).
One way is the nuclear norm, which is the sum of singular values.
It's often used in low rank matrix completion, such as in "Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization" by B. Recht et al.