Suppose the matrix F can be decomposed into orthogonal and symmetric parts $F = RU$ where R is the orthogonal matrix and U the symmetric one.
If I suppose that $F$ is symmetric by default does that imply that $F = U $ ? I tried to prove that but I am stuck here:
$$ F^T = (RU)^T = UR^T= F$$ $$ RU = UR^T $$ $$ RUR = U$$
But I am not able to extract any useful conclusion from the last equation. Thanks.
EDIT: if F=U for symmetric F is not generally true, then on what conditions would it be true? I have solved a few exercises where they are the same...