# matrix monotone

A real function $f$ on a real interval $I$ is said to be matrix monotone of $n$, if

 $A\leq_{L}B\Rightarrow f(A)\leq f(B)$ (1)

for all Hermitian $n\times n$ matrices $A,B$ with spectra contained in $I$. Here $\leq_{L}$ denotes the Loewner order, and the notation $f(A)$ is explained in the entry functional calculus for Hermitian matrices.

Title matrix monotone MatrixMonotone 2013-03-22 13:34:57 2013-03-22 13:34:57 mathcam (2727) mathcam (2727) 12 mathcam (2727) Definition msc 40A30 ALeqBForHermitianMatricesAB OperatorMonotone