Empty Matrices in MATLAB

What matrix has zero norm, unit determinant, and is its own inverse? The conventional answer would be that there is no such matrix. But the empty matrix [ ] in MATLAB satisfies these conditions:

>> A = []; norm(A), det(A), inv(A)
ans =
     0
ans =
     1
ans =
     []

While many MATLAB users will be familiar with the use of [ ] as a way of removing a row or column of a matrix (e.g., A(:,1) = []), or omitting an argument in a function call (e.g., max(A,[],2)), fewer will be aware that [ ] is just one in a whole family of empty matrices. Indeed [ ] is the 0-by-0 empty matrix

>> size([])
ans =
     0     0

Empty matrices can have dimension n-by-0 or 0-by-n for any nonnegative integer n. One way to construct them is with double.empty (or the empty method of any other MATLAB class):

>> double.empty
ans =
     []
>> double.empty(4,0)
ans =
   Empty matrix: 4-by-0

What makes empty matrices particularly useful is that they satisfy natural generalizations of the rules of matrix algebra. In particular, matrix multiplicaton is defined whenever the inner dimensions match up.

>> A = double.empty(0,5)*double.empty(5,0)
A =
     []
>> A = double.empty(2,0)*double.empty(0,4)
A =
     0     0     0     0
     0     0     0     0

As the second example shows, the product of empty matrices with positive outer dimensions has zero entries. This ensures that expressions like the following work as we would hope:

>> p = 0; A = ones(3,2); B = ones(3,p); C = ones(2,3); D = ones(p,3);
>> [A B]*[C; D]
ans =
     2     2     2
     2     2     2
     2     2     2

In examples such as this empty matrices are very convenient, as they avoid us having to program around edge cases.

Empty matrices have been in MATLAB since 1986, their inclusion having been suggested by Rod Smart and Rob Schreiber . A 1989 MATLAB manual says

We’re not sure we’ve done it correctly, or even consistently, but we have found the idea useful.

In those days there was only one empty matrix, the 0-by-0 matrix, and this led Nett and Haddad (1993) to describe the MATLAB implementation of the empty matrix concept as “neither correct, consistent, or useful, at least not for system-theoretic applications”. Nowadays MATLAB gets it right and indeed it adheres to the rules suggested by those authors and by de Boor (1990). If you are wondering how the values for norm([]), det([]) and inv([]) given above are obtained, see de Boor’s article for an explanation in terms of linear algebra transformations.

The concept of empty matrices dates back to before MATLAB. The earliest reference I am aware of is a 1970 book by Stoer and Witzgall. As the extract below shows, these authors recognized the need to support empty matrices of varying dimension and they understood how multiplication of empty matrices should work.