= Operator monotonicity of logarithm
For positive-definite <Hermitian matrices>, the <matrix logarithm> preserves the <Loewner order>:
$$
0<A\leq B\quad\Longrightarrow\quad\log A\leq\log B.
$$
No commutativity is required. With the natural <logarithm>, the <spectral theorem> gives the convergent integral representation
$$
\log A=\int_0^\infty\left[\frac{I}{1+t}-(A+tI)^{-1}\right]dt.
$$
If $0<C\leq E$, then $E^{-1/2}CE^{-1/2}\leq I$; taking its inverse and applying the reverse congruence gives $C^{-1}\geq E^{-1}$. Hence
$$
\log B-\log A=\int_0^\infty[(A+tI)^{-1}-(B+tI)^{-1}]\,dt\geq0.
$$
This proves <operator monotonicity> by a resolvent comparison.
For singular <positive semidefinite matrices>, the ordinary finite <matrix logarithm> is not defined at a zero <eigenvalue>. If $A$ and $B$ have the same <kernel of a linear map>, compress to its orthogonal complement and apply the theorem there. More generally $\log(A+\varepsilon I)\leq\log(B+\varepsilon I)$ holds for every $\varepsilon>0$, but divergent logarithms cannot simply be replaced by finite operators on different supports. The spectral definition and this support qualification are essential.<The Theory of Quantum Information by John Watrous/Mathematical preliminaries>[§1.1.2, equation(1.149), printed p28, spectral definition for positive-definite operators]{ref}
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