Lanchester's Square Law states that given two armies, $x$ and $y$, with the army units' relative strengths $\alpha$ and $\beta$, respectively, we can write two differential equations for the sizes of the armies as a function of time: $$\dot{x}=-\beta y,$$ $$\dot{y}=-\alpha x.$$ This can be solved analytically for the time functions $x(t)$ and $y(t)$ with the appropriate starting conditions, but more importantly, you can formally divide the two equations and, after integration, get $$\alpha x^2-\beta y^2=\alpha x_0^2-\beta y_0^2$$ throughout the whole battle. This means the strength of an army can be characterized by the product of its relative unit strength and the square of its size, because if this value is greater for one army at the beginning than that of the other army, this difference will stay constant until the weaker army's size eventually reaches zero.
Now if both armies consist of different unit types, components, and each component's units can have different strengths against other components, we can summarize all the components in a vector $\mathbf{x}$, and all the relative strengths in a matrix $\mathbf{A}$, where all the elements of $\mathbf{A}$ are negative or zero, and write a matrix differential equation: $$\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}.$$ This, again, can be solved in a number of (rather tedious) ways for $\mathbf{x}(t)$, but I would be interested if there is an "army strength" expression for this generalized case as well, which lets you easily know which component dies first.
So basically what I'm looking for, is a scalar function of the initial component sizes and relative strenghts which describes the strength of a component. That is, if said scalar value is greater for one component than the other, it will die later than one with a smaller value.
My initial thought was that a function $f(\mathbf{x})$ is needed, whose value is constant throughout the battle, just like $\alpha x^2-\beta y^2$ was.
It would require $$\frac{df}{dt}=\frac{df}{\mathbf{dx}}\frac{\mathbf{dx}}{dt}=0,$$ which could be written as $$\frac{df}{\mathbf{dx}}\mathbf{A}\mathbf{x}=0.$$ Unfortunately, that's about where my math skills extend to, so I don't really know where to go from here. I don't know how such an equation could be solved for $f$. What I do know is that $\mathbf{x}$ in general is obviously not a null vector, and $\det{\mathbf{A}}\neq0$ in general, and of course $\frac{df}{\mathbf{dx}}=\mathbf{0}$ wouldn't make much sense as a solution.
So my first question is, how could this equation be solved to provide some information about $f$? Secondly, is it a good way to my goal at all, and if not, what other ways could there be (if at all), to easily say which component of the armies will die first, and what the values remaining for the other components will be at that time, without having to calculate the exact time funcions.
Thanks for the answers!