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.



information measure for matrix that is analogous to rank
The notion of [stable rank](https://nickhar.wordpress.com/2012/02/29/lecture-15-low-rank-approximation-of-matrices/) is often used in the low-rank matrix approximation literature to offer a more tractable surrogate for rank. This does not necessarily satisfy all your requirements, but might be still suitable for your purposes.


