In Mathematica it is possible to apply arbitrary functions to square matrices (not element-wise, but using matrix rules).
This seemingly works with Sign function: MatrixFunction[Sign, A]. For instance, this method gives
$\text{sign}\left( \begin{array}{cc} 1 & -8 \\ 1 & 7 \\ \end{array} \right)=1$
(e.g., produces an identity matrix) but
$\text{sign}\left( \begin{array}{cc} 1 & -8 \\ -1 & 7 \\ \end{array} \right)=\left( \begin{array}{cc} -\frac{3}{\sqrt{17}} & -\frac{8}{\sqrt{17}} \\ -\frac{1}{\sqrt{17}} & \frac{3}{\sqrt{17}} \\ \end{array} \right)$
The function can be even applied to some zero divisors.
I wonder, what can be the meaning and properties of sign function, applied to matrices this way?
Can the matrices that have sign equal to $1$ be considered "positive" in some sense, even if they have negative elements?
I also noticed that when applied to split-complex numbers in matrix form it takes 9 possible values: $0,1,-1, j, -j, 1/2+j/2,1/2-j/2, -1/2+j/2, -1/2-j/2$. This is in contrast to complex numbers, where the set of values is infinite.
When applied to dual numbers, it gives 5 different values.
The usual rule $\text{sign } (AB)=\text{sign }A\cdot \text{sign } B$ still holds though.
Can a space be categorized into how many values the sign function can take there, into infinitely-signed and finitely-signed?
I'm not sure what Mathematica is doing here, but if it was Maple it would be applying the holomorphic functional calculus. This defines $f(A)$ whenever $f$ is analytic in a neighbourhood of the eigenvalues of matrix $A$. If $D$ is a diagonal matrix, $f(D)$ is diagonal with diagonal elements the values of $f$ on the diagonal elements of $D$, and $f(S D S^{-1}) = S f(D) S^{-1}$. In particular, if this is what is being used, $f(A)$ will be the identity matrix if the eigenvalues of $A$ are positive reals.