Today I Learned

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

48: Self-Adjoint Linear Operators

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.

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