How to expand a symbolic matrix expression?

I want to expand all terms in a complex symbolic matrix expression by the distributive law of the expression.

tempa1 = symmatrix('A1',[2 2]);
tempb1 = symmatrix('B1',[2 2]);
tempc1 = symmatrix('C1',[2 2]);
tempd1 = symmatrix('D1',[2 2]);

for nn = 2:5

    tempa2 = symmatrix(sprintf('A%d',nn),[2 2]);
    tempb2 = symmatrix(sprintf('B%d',nn),[2 2]);
    tempc2 = symmatrix(sprintf('C%d',nn),[2 2]);
    tempd2 = symmatrix(sprintf('D%d',nn),[2 2]);

    tempa = tempa1*tempb2+tempc1*tempd2+tempb1*tempa2+tempd1;
    tempb = tempa2*tempb1+tempc2*tempd1+tempb2*tempa1+tempd2;
    tempc = tempa1*tempb1+tempc1*tempd1+tempb1*tempa1+tempd2;
    tempd = tempa2*tempb2+tempc2*tempd2+tempb2*tempa2+tempd1;

    tempa1 = tempa;
    tempb1 = tempb;
    tempc1 = tempc;
    tempd1 = tempd;

end

vec_1 = symmatrix('vec_1',[2 1]);

result = tempa*vec_1;

When the variable result is executed, it is output as a complex matrix expression.

The variable result is a 2 by 2 symbolic matrix, and each element is also returned as a symbolic expression.

However, the information I need is not an element expression, but a string of expressions output when the variable result is executed.

When the variable result is executed, looking at the output result, the variable vec_1 is expressed by multiplying the result of tempa outside the bracket.

I want all the brackets to be solved and each term to be multiplied by vec_1.

A bulti-in function called 'expand' is identified as the most similar function to what I want.

However, this function is not supported for symmatrix input.

To use this function, if you run symmatrix2sym on the variable result, it returns all element representations.

Please, let me know that.

Thank you.

[Dyuman Joshi]

If it not necessary to use symmatrix, you can define 2x2 symbolic variables instead.

[Lee Jong Hyun]

Thank you for your advise.

The reason for using symmatrix is to apply non-commutativity.

[Paul]

Hi Lee Jong Hyun,

I could be wrong, but I don't think you'll be able to easily get the result you're looking for.

But suppose you could, i.e., get an expression for tempa in the form (n = 2)

syms A1 A2 B1 B2 C1 D1 D2 [2 2] matrix
syms vec_1 [2 1] matrix
tempa = A1*B2*vec_1 + B1*A2*vec_1 + C1*D2*vec_1 + D1*vec_1

tempa = 

What whould be the next step in the process? Maybe there's a way to achieve the goal without forming tempa as shown?

Answer 1

[Star Strider]

I am not certain what you intend by ‘expand’.

One option could be:

sympref('AbbreviateOutput',false);

that I added, and another the expand function (that does not work here). If you want to see the elements of the matrices, you would need to define their elements, as described in Convert Numeric Matrix to Symbolic Matrix Variable.

sympref('AbbreviateOutput',false);
tempa1 = symmatrix('A1',[2 2]);
tempb1 = symmatrix('B1',[2 2]);
tempc1 = symmatrix('C1',[2 2]);
tempd1 = symmatrix('D1',[2 2]);

for nn = 2:5

    tempa2 = symmatrix(sprintf('A%d',nn),[2 2]);
    tempb2 = symmatrix(sprintf('B%d',nn),[2 2]);
    tempc2 = symmatrix(sprintf('C%d',nn),[2 2]);
    tempd2 = symmatrix(sprintf('D%d',nn),[2 2]);

    tempa = tempa1*tempb2+tempc1*tempd2+tempb1*tempa2+tempd1;
    tempb = tempa2*tempb1+tempc2*tempd1+tempb2*tempa1+tempd2;
    tempc = tempa1*tempb1+tempc1*tempd1+tempb1*tempa1+tempd2;
    tempd = tempa2*tempb2+tempc2*tempd2+tempb2*tempa2+tempd1;

    tempa1 = tempa;
    tempb1 = tempb;
    tempc1 = tempc;
    tempd1 = tempd;

end

vec_1 = symmatrix('vec_1',[2 1]);

result = tempa*vec_1

result = 

.

[Lee Jong Hyun]

The intention of using 'expand' is to return a string of expressions expressed by multiplying all terms by a vector.

I'm not curious about the elements of the matrix. As in your answer, I want to get the string of expressions with matrices A,B, not elemental expressions.

[Lee Jong Hyun]

For a simple example,

tempa1 = symmatrix('A1',[2 2]);
tempb1 = symmatrix('B1',[2 2]);
tempc1 = symmatrix('C1',[2 2]);
tempd1 = symmatrix('D1',[2 2]);
for nn = 2:3
    tempa2 = symmatrix(sprintf('A%d',nn),[2 2]);
    tempb2 = symmatrix(sprintf('B%d',nn),[2 2]);
    tempc2 = symmatrix(sprintf('C%d',nn),[2 2]);
    tempd2 = symmatrix(sprintf('D%d',nn),[2 2]);
    tempa = tempa1*tempb2+tempc1*tempd2+tempb1*tempa2+tempd1;
    tempb = tempa2*tempb1+tempc2*tempd1+tempb2*tempa1+tempd2;
    tempc = tempa1*tempb1+tempc1*tempd1+tempb1*tempa1+tempd2;
    tempd = tempa2*tempb2+tempc2*tempd2+tempb2*tempa2+tempd1;
    tempa1 = tempa;
    tempb1 = tempb;
    tempc1 = tempc;
    tempd1 = tempd;
end
vec_1 = symmatrix('vec_1',[2 1]);
result = tempa*vec_1

result = 

In the above result, the returned output is follows.

(A2*B2 + B2*A2 + C2*D2 + (A2*B1 + B2*A1 + C2*D1 + D2)*A3 + (A1*B2 + B1*A2 + C1*D2 + D1)*B3 + (A1*B1 + B1*A1 + C1*D1 + D2)*D3 + D1)*vec_1

But the result I want is follows.

A2*B2*vec_1 + B2*A2*vec_1 + C2*D2*vec_1 + A2*B1*A3*vec_1 + B2*A1*A3*vec_1 + C2*D1*A3*vec_1 + D2*A3*vec_1 + A1*B2*B3*vec_1 + B1*A2*B3*vec_1 + C1*D2*B3*vec_1 + D1*B3*vec_1 + A1*B1*D3*vec_1 + B1*A1*D3*vec_1 + C1*D1*D3*vec_1 + D2*D3*vec_1 + D1*vec_1

[Star Strider]

I am not certain how to get that result, since the functions I would normally use do not work on symmatrix expressions.

[Dyuman Joshi]

It might not be possible to get the result OP wants to obtain (as mentioned above) because one can not manipulate the result obtained from the code.

These are the all functions that accept symbolic matrix variables as input, the only functions that manipulates an expression are formula and displayFormula, both of which give the same output.

methods symmatrix

Methods for class symmatrix:

adjoint        csc            disp           exp            isempty        latex          minus          ne             pretty         size           tan            uplus          
cat            ctranspose     display        formula        isequal        ldivide        mldivide       norm           rdivide        subs           times          vertcat        
conj           curl           divergence     gradient       isequaln       length         mpower         numel          rhs            sym            trace          vpa            
cos            det            double         horzcat        kron           lhs            mrdivide       plus           sec            symmatrix      transpose      
cot            diff           eq             inv            laplacian      log            mtimes         power          sin            symmatrix2sym  uminus         

Static methods:

empty