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Monotonic Functions
Real Positive Semidefinite Matrix Order
Matrix Scalar Product
Real Inner Product Norms
Real Matrix Inverses
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Monotonic Functions of Real Matrices


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.


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..


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 $f(A) = \tr AB$.


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

  1. Future editions will include a proof. ↩︎
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