I know from a theorem that every hermitian and skew-hermitian matrix is similar to a diagonal matrix.
But, is this fact also true for symmetric and skew-symmetric matrices?
And, symmetric matrices have real eigenvalues, what about symmetric matrices that have complex entries?
Best Answer
This is just an "add-on" for the complex symmetric case.
No, complex symmetric matrices do not need to be diagonalizable. Consider $$ \pmatrix{1 & i\\ i & -1}, $$ which is symmetric but is not diagonalisable.
However, for any complex symmetric matrix $A$, there is a unitary matrix $U$ such that $A=UDU^T$, where $D$ is a nonnegative diagonal matrix (note that $^T$ stands here for the usual transposition, which is not same as the conjugate transpose usually seen in the context of complex matrices). This is referred to as the Takagi's factorization.