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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($I_2$).

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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.

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  • $\begingroup$ sum signular values of $A(1)$ is 3,(2+1+0) same as for $A(0)$ 3 = 1 + 1 +1? $\endgroup$ – ahala Jun 16 '16 at 18:07
  • $\begingroup$ Its a commonly used surrogate for rank which is continuous and even convex, but will not necessarily give you the same result as rank (even on your example). $\endgroup$ – Batman Jun 16 '16 at 18:10
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This seems close: $$ I(A(a)) = \frac{n^2}{||A(a)||^2_F}$$, where $||A(a)||_F$ is the matrix entrywise 2-norm. For the example in the question: $ I(A(0)) = 3$ and $ I(A(1)) = 9/5 = 1.8$

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