Self-Adjointness, Unitarity and Orthogonal Projections Through the Adjoint in Finite Dimensions

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space that has an orthonormal basis $e\in V^{n}$ for some natural number $n$, where $V^{n}$ is the set of $n$-tuples in $V$.

By Uniqueness of the Adjoint, and Existence in Finite Dimensions every linear operator on $V$ has exactly one adjoint, which is again a linear operator on $V$; it is written with a star. Products of operators and the identity operator $\mathrm{id}_{V}$ are as in that definition.

Let $T$ and $P$ be linear operators on $V$. Then the following hold.

1. (Self-adjointness) $T$ is self-adjoint if and only if $T^{*}=T$.

2. (Unitarity) $T$ is unitary if and only if $T^{*}T=\mathrm{id}_{V}$ and $TT^{*}=\mathrm{id}_{V}$.

3. (Orthogonal projections) $P$ is an orthogonal projection if and only if $PP=P$ and $P^{*}=P$.