Help with symbolic math and matrix multiplication

The following code should prove with symbolic math this well-known statement: for every A in this form, given a matrix U as described in the code, U'AU is diagonal. However there is some numerical problem with the conjugate operation. Anybody can help?

syms A11 A12 A13 A14 A15;
A=[A11 A12 A13 A14 A15 A14 A13 A12;
   A12 A11 A12 A13 A14 A15 A14 A13;
   A13 A12 A11 A12 A13 A14 A15 A14;
   A14 A13 A12 A11 A12 A13 A14 A15;
   A15 A14 A13 A12 A11 A12 A13 A14;
   A14 A15 A14 A13 A12 A11 A12 A13;
   A13 A14 A15 A14 A13 A12 A11 A12;
   A12 A13 A14 A15 A14 A13 A12 A11;];

g=sym('exp(1i*2*pi/8)');

   U1=sym(zeros(8,8));

   for i=1:size(U1,1)
       for j=1:size(U1,2)
           U1(i,j)=g^mod((i-1)*(j-1),8);
       end
   end

D=U1'*A*U1/8;

[Walter Roberson]

syms A11 A12 A13 A14 A15;
A=[A11 A12 A13 A14 A15 A14 A13 A12;
   A12 A11 A12 A13 A14 A15 A14 A13;
   A13 A12 A11 A12 A13 A14 A15 A14;
   A14 A13 A12 A11 A12 A13 A14 A15;
   A15 A14 A13 A12 A11 A12 A13 A14;
   A14 A15 A14 A13 A12 A11 A12 A13;
   A13 A14 A15 A14 A13 A12 A11 A12;
   A12 A13 A14 A15 A14 A13 A12 A11;];

g = exp(1i*2*sym(pi)/8)

g = 

U1 = zeros(8,8, 'sym');

for i=1:size(U1,1)
    for j=1:size(U1,2)
        U1(i,j) = g^mod((i-1)*(j-1),8);
    end
end

D = U1'*A*U1/8

D = 

Looks diagonal to me.

Answer 1

[Karan Singh]

Hi @riccardo stara,

I dont know when the entries are created from string expressions (e.g. sym('exp(1i2pi/8)')) the symbolic engine may not recognize that the expression represents a complex number on the unit circle. I have tried the other way round. I define the complex constant using MATLAB’s numeric conversion. Simliar to Walter's code. Also the the symbols in your matrix are meant to be real, you can declare them as real.

g = exp(sym(1i)*2*pi/8);

This is my code

syms A11 A12 A13 A14 A15 real;

A = [A11, A12, A13, A14, A15, A14, A13, A12;
     A12, A11, A12, A13, A14, A15, A14, A13;
     A13, A12, A11, A12, A13, A14, A15, A14;
     A14, A13, A12, A11, A12, A13, A14, A15;
     A15, A14, A13, A12, A11, A12, A13, A14;
     A14, A15, A14, A13, A12, A11, A12, A13;
     A13, A14, A15, A14, A13, A12, A11, A12;
     A12, A13, A14, A15, A14, A13, A12, A11];

g = exp(sym(1i)*2*pi/8);

U1 = sym(zeros(8,8));
for i = 1:8
    for j = 1:8
        U1(i,j) = g^mod((i-1)*(j-1), 8);
    end
end

D = simplify(U1' * A * U1 / 8)

D = 

Karan