# matrix block notation ambiguity

In block notation, the matrix $$\begin{pmatrix} 1&2&3&4 \\ 5&6&7&8 \\ a&b&c&d \\ e&f&g&h\end{pmatrix}$$ can be writen as $$\begin{pmatrix} A_1& A_2 \\ A_3&A_4\end{pmatrix}$$ where $$A_1 = \begin{pmatrix} 1&2 \\ 5&6\end{pmatrix}$$, $$A_2 = \begin{pmatrix} 3&4 \\7&8\end{pmatrix}$$, ....

I would like to write this matrix $$\begin{pmatrix}\begin{pmatrix} 1&2 \\ 5&6 \end{pmatrix}&\begin{pmatrix} 3&4 \\ 5&6 \end{pmatrix}\\\begin{pmatrix} a&b \\ e&f \end{pmatrix}&\begin{pmatrix} c&d\\ g&h\end{pmatrix}\end{pmatrix}$$ as $$\begin{pmatrix} A_1& A_2 \\ A_3&A_4\end{pmatrix}$$, but this looks like block notation. How do you handle this ambiguity?

The approach I think I will take is that block notation will allways be indicated by lines $$\left( \begin{array}{c|c} A_1& A_2 \\\hline A_3&A_4\end{array} \right)$$.

If the notation at the top is called block notation, what do you call the other notation? It is similar to a tensor product, but I don't think it actually is.

• Your notation means that the entries of the matrix are matrices. So you write down an element of $M_n(R)$ for the ring $R=M_m(K)$. This is not block notation, but just matrix notation with matrix entries from $R=M_m(K)$. Commented May 30, 2023 at 18:20
• Thanks for input. How do I distinguish the two notations? How do I tell the reader, this is block notation and this over here is matrix notation? Commented May 30, 2023 at 20:29
• Are you ever in a situation where the two notations are being used near each other, and it matters very much which one is which? (Because, for example, as vector spaces, who cares?) Is there a reason you can't just write in words "where in this case blah means blah" in a couple of places?
– JBL
Commented Jun 2, 2023 at 21:14
• @JBL good point. It seems there is no well established convention for distinguishing these two uses of the notation. So, I'll have to use a work-around like you suggest. Commented Jun 3, 2023 at 16:07