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 $I(A(0)) = 3$ and $I(A(1)) = 2$? Rank is not continuous, Shannon information entropy on eigenvalues does not give desired values.