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

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  • $\begingroup$ en.wikipedia.org/wiki/Von_Neumann_entropy $\endgroup$
    – Steve Huntsman
    Jun 16, 2016 at 19:25
  • $\begingroup$ $S(\rho) = -tr(\rho\log(\rho))$ equals zero for any identity matrix while the dimension information lost; and $S(A(1))$ from my example does not reduce to S($I_2$). $\endgroup$
    – ahala
    Jun 16, 2016 at 19:58
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    $\begingroup$ If I have not made a silly mistake, rank($A$) =rank($A^*A$) by some version of the polar decomposition or SVD. So any function on $[0,\infty)^n$ that is a good substitute for "count the number of non-zero entries" should give, when applied to the diagonalization of $A*A$, give some substitute for rank which might fit your purposes $\endgroup$
    – Yemon Choi
    Jun 16, 2016 at 22:22

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The notion of stable rank 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.

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