Is $\sqrt{1+x^2}$ matrix monotone? A function $f(x)$ is matrix monotone if $f(A)-f(B)$ is positive semidefinite whenever $A-B$ is positive semidefinite for positive semidefinite matrices $A, B$. Is $\sqrt{1+x^2}$ matrix monotone?

I think yes.  My general thinking is that the positive definite property is tied
to a matrix's determinate being non-zero, and that the square of a matrix will merely
amplify the magnitude of the determinate, but not zero it out.
I emphasize that this is strictly a guess.
