Symmetric Matrix Product is not Symmetric

Question

When we have two symmetric matrices x and y, the matrix z = xyx is theoretically also symmetric. However this is not exactly true in Matlab:

x = randn(3);
y = randn(3);
x = x*x';
y = x*x';
z = x*y*x;
issymetric(z)

Why does this happen and what can I do about it? Since I do not want to do

z = .5*(z+z')

all anwers in this stackoverflow question are unsatisfactionary.

Answer 1

The problem is that Matlab does not perform the calculations exact, and thus the finite precision of the floating point calculations introduces some truncation errors.

If you run the example with symbolic math (no truncation, exact), you will see that z is actually symmetric.

x = sym(randn(3));
y = sym(randn(3));
x = x*x';
y = y*y';
z = x*y*x;

% issymmetric does not take symbolic expressions as argument, convert to double
issymmetric(double(z))

Since you now do the double conversion at the end, and not in the intermediate steps, the matrix will remain symmetric.