I don't understand the difference between symmetry and hermitian / skew-symmetry and skew-
hermitian.
In my book, it says that difference between these two is whether euclidean space is real or complex.
In complex euclidean space, T is called hermitian if (T(x).y)=(x.T(y))
and If T is hermitian, then eigenvalue is real.
but in complex euclidean space, inner product is complex number.
How can (T(x).x)=(x.T(x)) be true without conjugate sign if T is hermitian?
And why eigenvalue is real in hermitian transformation?
Shouldn't inner product be complex number?
Then eigenvalue= (T(x).x)/(x.x) we know (x.x) is real and (T(x).x) is complex number.
How can eigenvalue be real??
A real matrix is symmetric if $A = A^T$. A complex matrix $A$ is hermitian if $A = \overline{A}^T$. Similarly we define a skew-symmetric matrix and skew-hermitian matrix. Consider $T: \Bbb C^2 \to \Bbb C^2$ given by $T(w,z)=(iw,iz)$. Then $i \in \Bbb C$ is an eigenvalue of $T$, and every vector is an eigenvector.
Remember that in a complex vector space, we must have: $$\langle {\bf x}, {\bf y}\rangle = \overline{\langle {\bf y}, {\bf x} \rangle}.$$ $T$ being a hermitian operator means that: $$\langle T{\bf x}, {\bf y} \rangle = \langle {\bf x}, T{\bf y}\rangle,$$ and that takes into account that first property I said.