# What does $R(G:X)$ mean in Rao's book?

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

• 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. Commented Jan 17, 2023 at 0:54
• @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? Commented Jan 17, 2023 at 1:19

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.
• I am still not clear if $G:X$ means $[G,X]$ or $\begin{pmatrix} G &X \\ X^\intercal &0\end{pmatrix}.$ Commented Jan 17, 2023 at 15:57
• $\mathbf{G}:\mathbf{X}$ means $[\mathbf{G},\mathbf{X}].$ Commented Jan 17, 2023 at 15:59
• 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. Commented Jan 17, 2023 at 16:06