How to determine a function of a matrix is increasing or decreasing We know that  the derivative of a function can be used to determine whether the function is increasing or decreasing on any intervals in its domain. If $f'(x) > 0 $ at certain interval I, then the function is increasing on I; and, if $f'(x) < 0$  then the function is decreasing. 
My question is about functions of a matrix, rather than a scalar. Let us replace the scalar $x$ with a matrix $X$. Can we simply say that  the above statements are valid for $f(X)$ if we replace "<" and ">"  with "$\prec$" and "$\succ$", respectively, where  "$\prec$" represents "less than or equal" in to in the positive semi-definite partial ordering? 
Finally, here is an specific example:
 $$f(X)= \log \left|X + K_1\right|- \alpha \log \left|X + K_2\right|$$
where $X$, $K_1$, and $K_2$ are all symmetric positive semi-definite matrices, $\alpha$ is a scalar, and $|\cdot|$ represents the determinant. I'd like to determine when this function is increasing or decreasing. 

In order to define an increasing function $f$ between set $A$ and set $B$, you just need to have orders defined on those sets. So using positive semi-definite partial ordering on set $A$ is fine.
To go straight to the point, you don't need the orders to be total to define increasing (or more generally monotonic) functions. It is possible to define monotonic functions between partially ordered sets. But obviously the derivative can't be used as in the real case to decide monotonicity.
Regarding your specific function $f$ I doubt that it can monotonic as the real function $$g(x)=\log(x+a) - \alpha \log(x+b)$$ is already not monotonic as you can see considering its derivative $$g^\prime(x) = \frac{(1-\alpha)x+(b-\alpha a)}{(x+a)(x+b)}$$
