Does the Symbolic Toolbox support symbolic matrix operations?

I would like to define a variable and have the Symbolic Toolbox understand that it is a matrix and not a scalar.

Answer 1

[MathWorks Support Team]

To work with matrices in the Symbolic Toolbox, a variable must have each of its elements explicitly defined to be recognized as a matrix. Otherwise, variables are treated as scalars.

For example, this code snippet will define two symbolic matrices:

M = sym(zeros(2,2));

M1 = sym(zeros(2,2));

for row = 1:2;

   for col=1:2; 

     M(row,col) = sym(['a' num2str(row) num2str(col)]);

     M1(row,col) = sym(['b' num2str(row) num2str(col)]);

   end

end

These two matrices can now be multiplied together:

M*M1

 ans =

 [ a11*b11+a12*b21, a11*b12+a12*b22]

 [ a21*b11+a22*b21, a21*b12+a22*b22]

M1*M

 ans =

 [ a11*b11+a21*b12, b11*a12+b12*a22]

 [ b21*a11+b22*a21, a12*b21+a22*b22]

If your matrix elements are defined by an equation, use MESHGRID to help in defining that matrix. For example, the following code will create a symbolic matrix whose elements are described by the equation 1/(i+j-t), where "i" is the row index and "j" is the column index:

syms t

n = 3;

[J,I] = meshgrid(1:n);

A = sym(1./(I + J - t));

This produces

A

 A =

 [ 1/(2-t), 1/(3-t), 1/(4-t)]

 [ 1/(3-t), 1/(4-t), 1/(5-t)]

 [ 1/(4-t), 1/(5-t), 1/(6-t)]

[Walter Roberson]

The answer is NO, you cannot use sym or syms to define a variable as being a general matrix, you can only define it as being a matrix with a particular size and contents (possibly symbolic.) Operations then proceed on elements using the normal rules -- so for example A * B will only work if size(A,2) == size(B,1) . The results will not be in terms of A and B but will be in terms of the contents of the matrices.

The information that I find suggests that the symbolic engine itself, MuPAD, does not have a defined way of declaring that something is a matrix of unknown size, and does not have a defined way of doing (algebraic) matrix multiplication on two generic matrices -- only of doing multiplications upon the elements.

This is unlike (for example) Maple; Maple does have ways of creating non-commutative operations on generalized matrices.