Name for sum of diagonals for Hilbert Schmidt normalized matrix.

Question

Suppose that $A$ is an $n \times n$ matrix. Then the Hilbert-Schmidt norm of $A$ is given by $$ \vert \vert A \vert \vert_{HS}^2= \sum_{i,j=1}^n \vert{A_{i,j} \vert}^2. $$ Now, if I insist that $ \vert \vert A \vert \vert_{HS}^2 =1$ then the number $c(A)$ defined by $$ c(A)= \sum_{i=1}^n \vert{A_{i,i} \vert}^2 $$ is in $\lbrack 0,1 \rbrack$.

Does $c(A)$ have a name? Does it have any nice properties?

Answer 1

Use an unconstrained matrix $(U)$, the identity matrix $(I)$, the Hadamard product $(U\odot V)$ and the matrix inner product $\big(U:V={\rm tr}(U^TV)\big)$ to construct the constrained matrix $(A)$ and the scalar $(c)$ $$\eqalign{ A &= \frac{U}{\|U\|} \qquad\implies\quad A:A = \frac{U:U}{U:U} = 1 \\ c &= (I\odot A):(I\odot A) = \frac{(I\odot U):(I\odot U)}{U:U} \\ }$$ Since denominator selects all components of $U\,$ while the numerator selects only the diagonal components, the scalar function $c$ becomes a measure of the diagonal-ness of a given matrix.

In particular, note that

If $\,U\,$ is a diagonal matrix, then $c=1$

If $\,U\,$ has zeros along its diagonal, then $c=0$

Otherwise, $\;0\lt c\lt 1$