Does there exist a fairly standard notation for horizontal and vertical juxtaposition operations on matrices? (Vertical juxatposition can be called "stacking.")

For example, juxtaposing horizontally a matrix of the size $m\times n_1$ with a matrix of the size $m\times n_2$, one obtains a matrix of the size $m\times(n_1 + n_2)$. Stacking vertically a matrix $m_1\times n$ with a matrix $m_2\times n$, one obtains a matrix $(m_1 + m_2)\times n$.

I've seen the notation $$ A = [a_1|a_2|\dotsb|a_n] $$ for a matrix $A$ with columns $a_1,a_2,\dotsc,a_n$. It looks a bit ad hoc, unless we define $|$ as the horizontal juxtaposition operation that can be applied to any pair of matrices with the same number of rows. A notation for vertical juxtaposition is also needed.

Such notation would be quite useful for writing matrices defined by blocks or decomposing matrices into blocks (into rows or columns in particular).


Just draw a matrix of matrices:


  • $\begingroup$ This would be a matrix of matrices $2\times 2$, not a matrix with $4$ blocks. $\endgroup$ – Alexey Mar 20 '17 at 13:07
  • $\begingroup$ I would like to have an explicit notation (which could be used with a computer algebra system, for example), not a notation that needs to be accompanied with oral explanation or text in parentheses. $\endgroup$ – Alexey Mar 20 '17 at 13:12
  • $\begingroup$ So you're saying a $2\times 2$ matrix whose entries are $n\times n$ matrices is different from a $2n\times 2n$ matrix? I suppose you could say that one is a linear endomorphism on $R^n\oplus R^n$ whereas the other is a linear endomorphism on $R^{2n}$. But these two things are so clearly isomorphic that no one will be confused, even without an accompanying explanation. $\endgroup$ – Oscar Cunningham Mar 20 '17 at 13:22
  • $\begingroup$ For what it's worth, octave (and I presume MATLAB) thinks that "A=[1 2; 3 4]; B=[A A; A A]" is valid code which gives the answer you'd expect. $\endgroup$ – Oscar Cunningham Mar 20 '17 at 13:24

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