Programming and Data Types

Empty Matrices

A matrix having at least one dimension equal to zero is called an empty matrix. The simplest empty matrix is 0-by-0 in size. Examples of more complex matrices are those of dimension 0-by-5 or 10-by-0-by-20.

To create a 0-by-0 matrix, use the square bracket operators with no value specified.

• A = [];
  
  whos A
    Name      Size         Bytes  Class
  
    A         0x0              0  double array
  

You can create empty arrays of other sizes using the zeros, ones, rand, or eye functions. To create a 0-by-5 matrix, for example, use

• E = zeros(0,5)
  

Operating on an Empty Matrix

The basic model for empty matrices is that any operation that is defined for m-by-n matrices, and that produces a result whose dimension is some function of m and n, should still be allowed when m or n is zero. The size of the result should be that same function, evaluated at zero.

For example, horizontal concatenation

• C = [A B]
  

requires that A and B have the same number of rows. So if A is m-by-n and B is m-by-p, then C is m-by-(n+p). This is still true if m or n or p is zero.

Many operations in MATLAB produce row vectors or column vectors. It is possible for the result to be the empty row vector

• r = zeros(1,0)
  

or the empty column vector

• C = zeros(0,1)
  

As with all matrices in MATLAB, you must follow the rules concerning compatible dimensions. In the following example, an attempt to add a 1-by-3 matrix to a 0-by-3 empty matrix results in an error.

• [1 2 3] + ones(0,3)
  ??? Error using ==> +
  Matrix dimensions must agree.
  

Some MATLAB functions, like sum and max, are reductions. For matrix arguments, these functions produce vector results; for vector arguments they produce scalar results. Empty inputs produce the following results with these functions:

• sum([ ]) is 0
• prod([ ]) is 1
• max([ ]) is [ ]
• min([ ]) is [ ]

Using Empty Matrices with If or While

When the expression part of an if or while statement reduces to an empty matrix, MATLAB evaluates the expression as being false. The following example executes statement S0, because A is an empty array.

• A = ones(25,0,4);
  
  if A
     S1
  else
     S0
  end
  

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