# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

A linear operator $T: V \rightarrow V$ is self-adjoint iff it is its own adjoint, i.e. iff
$\langle T(x), y \rangle = \langle x, T(y) \rangle \quad \forall x, y \in V$.
This is equivalent to the condition that the matrix of $T$ with respect to any orthonormal basis is Hermitian (the matrix is its own conjugate transpose).
In addition, if $T$ is self-adjoint, then there exists an orthonormal eigenbasis $\beta$ for $V$ such that the matrix representation of $T$ with respect to $\beta$ is a diagonal matrix with real entries.