# Proving that the Composition of Two Self-Adjoint Operators is Self-Adjoint

1. Homework Statement
Prove or give a counterexample: the product of any two self-adjoint operators on a finite-dimensional inner-product space is self-adjoint.

2. Homework Equations
The only two equations I've used so far are:
$$\left\langle T v, w\right\rangle = \left\langle v, T^{*}w\right\rangle$$
and
$$\left(T \circ S \right)^{*} = S^{*} \circ T^{*}$$

3. The Attempt at a Solution
I started with the definition of adjoint, using $$T \circ S$$ :
$$\left\langle \left(T \circ S \right) v, w \right\rangle = \left\langle v, \left(T \circ S \right)^{*} w \right\rangle$$
Using the second relevant equation, I got
$$\left\langle \left(T \circ S \right) v, w \right\rangle = \left\langle v, \left(S \circ T \right) w \right\rangle$$
So the composition is self-adjoint only if $$T$$ and $$S$$ commute. But I don't know where to go from there.
Will self-adjoint operators necessarily commute?

## Answers and Replies

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