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I was reading Rao's chapter 4 of his Linear Statistical Inference, 2nd ed. He uses the notation $R(G:X)$ in section 4.i (p. 294, formula (e), p. 296 formula (4i.1.21)) and that notation appears again in the last paragraph of p. 300. Unfortunately, Rao does not define the notation and I could not quite point out the meaning from the context. Does anyone know what it means?

Another book uses something similar $S(X:V)$ and they also do not define the notation, and again I am not quite sure what it means. (To be more explicit, they define $S$ to mean the image space or column space, and my best guess is that $X:V$ simply means the concatenation $[X, V]$ of the two matrices. Both expressions $X:V$ and $G:X$ appear in the context of the General Gauss Markov Model.)

I know this is a long shot, but if you know, I'd appreaciate if you let me know.

Regards, W

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  • $\begingroup$ R(X) is the rank of matrix X and G : X is as in the question. Rao uses a script M to mean the range space (M = manifold), not R. $\endgroup$
    – G. Grothendieck
    Commented Jan 17, 2023 at 0:54
  • $\begingroup$ @G.Grothendieck Yes, Rao uses $M$ for column space. $S$ is used by Takeuchi, Yanai and Mukherjee. So you are saying that $G:X$ simply means the matrix with columns of $G$ and then of $X,$ what I called the concatenation? $\endgroup$
    – William M.
    Commented Jan 17, 2023 at 1:19

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Rao has a knack of resorting to two notations while explaining stuffs involving rank throughout the very same treatise: $\operatorname{rank}(\cdot); ~R(\cdot):$ see sec. $\rm 1b.6, ~1b.7$ and as G. Grothendieck correctly noted, Rao uses $\mathscr M(\cdot)$ to denote the range space.

$\mathbf{G}:\mathbf{X}$ is a partitioned matrix. It is evident from the derivation he is doing: observe $(\rm 4i.1.21)$

$$R\begin{pmatrix}\mathbf G & \mathbf X\\ \mathbf X' & \mathbf 0\end{pmatrix} = R(\mathbf{G}:\mathbf{X}) + R(\mathbf X);$$ it is easy to show using elementary row transformations.

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  • $\begingroup$ I am still not clear if $G:X$ means $[G,X]$ or $\begin{pmatrix} G &X \\ X^\intercal &0\end{pmatrix}.$ $\endgroup$
    – William M.
    Commented Jan 17, 2023 at 15:57
  • $\begingroup$ $\mathbf{G}:\mathbf{X}$ means $[\mathbf{G},\mathbf{X}].$ $\endgroup$
    – User1865345
    Commented Jan 17, 2023 at 15:59
  • $\begingroup$ Okay, thanks. That was my guess all along, however, Rao uses $[G \mid X]$ to denote $[G, X]$ almost everywhere else in the book and $G:X$ only appears in the three spots I mentioned in the question without defining it ever. $\endgroup$
    – William M.
    Commented Jan 17, 2023 at 16:06
  • $\begingroup$ His writing is pretty enjoyable but thanks to these sort of notation changes, it can be itchy a bit. $\endgroup$
    – User1865345
    Commented Jan 17, 2023 at 16:09
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    $\begingroup$ Rao is not my first choice; it's for reference only. Sometimes, it takes time to comprehend but overall a good content provided you already are well-versed with what you are supposed to read. $\endgroup$
    – User1865345
    Commented Jan 17, 2023 at 16:14

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