How does this expression follow from Moore-Penrose inverse definition?

Question

I'm trying to understand a proof that uses a Moore-Penrose (pseudo)inverse. I don't have a very deep understanding of the M-P inverse, but I understand the four-property definition and basic properties that one can find in the Wikipedia page and other introductory sources. However, I can't see how to derive the expression below from them. No doubt the derivation has to do with particular properties involved in the proof (perhaps the role of diagonal matrices or stochastic vectors and matrices in it).

Here are the assumptions for the proof of a lemma on pp. 38-39 of Hartfiel's Markov Set-Chains, Springer 1998. I'm probably giving more detail than necessary, but I don't know what matters. (I am simplifying Hartfiel's notation slightly. His $K$ is my $K$ plus another matrix $F$.)

$X$ is the diagonal matrix for stochastic row vector $x$; $Y$ is the same for stochastic row vector $y$. $\alpha$ and $\beta$ are nonnegative numbers such that $\alpha + \beta = 1$. $A$ and $B$ are stochastic matrices (each row sums to 1) with elements $a_{ij}$ and $b_{ij}$, respectively.

Define matrix $K$ by

$$K = (\alpha X + \beta Y)^+ \,\, (\alpha XA + \beta YB)$$

where $^+$ indicates the Moore-Penrose inverse. Hartfiel claims that

$$k_{ij} = \frac{\alpha x_i a_{ij} + \beta y_i b_{ij}}{\alpha x_i + \beta y_i}$$

if $\alpha x_i + \beta y_i > 0$, or

$$k_{ij} = 0$$

if $\alpha x_i + \beta y_i = 0$.

This is a (to me) surprisingly simple expression. The numerator in the first case is just element $i,j$ of $(\alpha XA + \beta YB)$.

So in this case the effect of multiplying by the M-P inverse of $\alpha X + \beta Y$ is that each element of the resulting matrix is either zero is or simply the result of dividing each element of the right-hand matrix by the weighted sum of the corresponding vector elements. Why is that?

Is there an easy way to see how this follows from one of the Moore-Penrose inverse definitions? (Or do I need a lot more study to understand this expression?!?)

Answer 1

In general, if $D$ is a diagonal matrix and $C=D^+$, then $C$ is also a diagonal matrix with $$ c_{ii}=\begin{cases} d_{ii}^{-1}&\text{ when }d_{ii}\ne0,\\ 0&\text{ when }d_{ii}=0. \end{cases} $$ It is easy to verify that the $D^+$ defined this way satisfies the four defining properties of Moore-Penrose pseudo-inverse. If you permute the standard basis vectors of $\mathbb R^n$ (or $\mathbb C^n$), so that $D=D_1\oplus0$, then the result above is just saying that $D^+=D_1^{-1}\oplus0$. Essentially, it means the Moore-Penrose pseudo-inverse of an invertible matrix $D_1$ is just the usual inverse $D_1^{-1}$, and the M-P pseudo-inverse of the zero matrix is the zero matrix itself.