matrix monotone

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

$$A{\le }_{L}B\Rightarrow f(A)\le f(B)$$ (1)

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

Title	matrix monotone
Canonical name	MatrixMonotone
Date of creation	2013-03-22 13:34:57
Last modified on	2013-03-22 13:34:57
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	12
Author	mathcam (2727)
Entry type	Definition
Classification	msc 40A30
Related topic	ALeqBForHermitianMatricesAB
Related topic	OperatorMonotone