Some of the things I've learned every day since Oct 10, 2016
48: Self-Adjoint Linear Operators
November 27, 2016Posted by on
A linear operator is self-adjoint iff it is its own adjoint, i.e. iff
This is equivalent to the condition that the matrix of with respect to any orthonormal basis is Hermitian (the matrix is its own conjugate transpose).
In addition, if is self-adjoint, then there exists an orthonormal eigenbasis for such that the matrix representation of with respect to is a diagonal matrix with real entries.