Monotonic Functions of Real Matrices
Why
Since we have a partial order on the set of positive semidefinite matrices, we can study which familiar functions are have order-preserving or order-reversing properties.
Norms
It would be nice if the matrix norm induced by the matrix scalar produce (see Matrix Scalar Product) was an isotonic function. In other words, if $A, B \in \mathbfsf{S} ^d$ satisfy $A \geq B$, does $\norm{A} \geq \norm{B}$?
Since $\norm{A}^2 = \tr A^2$, we should study the trace first..
Trace
Let $f: \mathbfsf{S} ^d \to \R $ defined by
$f(A) = \tr A$.
In other words, the function $f$ is the restriction of the trace function onto the set of symmetric matrices.
Let $B \in \mathbfsf{S} ^d$
Let $f_B: \mathbfsf{S} ^d \to \R $ defined by
$f(A) = \tr AB$.
Inversion
Let $A \in \mathbfsf{S} _{++}^d$.
Then the map $f: \mathbfsf{S} _{++}^d \to
\mathbfsf{S} _{++}^d$ satisfying $f(A) =A^{-1}$ is
an isotonic function mapping the (open) positive
definite cone into itself.1
- Future editions will include a proof. ↩︎