Relationship between isometry, self-adjoint and positive

Question

I read "Linear Algebra Done Right" recently, and I wonder what is the relationship between isometry and self-adjoint operators on a finite dimensional complex space. It seems like their intersection includes involutory matrices. Also, a geometric understanding of a normal operator is that if $T\in L(V)$ is normal if and only if there exists some orthonormal coordinate system for $V$ such that $T$ fixes each coordinate axis. And if $T$ is an isometry, the scalar should be $1$. I wonder what are the geometric interpretations of a self-adjoint and even a positive operator.

Answer 1

Among the normal operators on a finite-dimensional complex inner product space, the isometries are distinguished by having eigenvalues all of which have absolute value 1. The self-adjoint operators are distinguished among the normal operators by having eigenvalues all of which are real. The positive operators are distinguished among the normal operators by having eigenvalues all of which are nonnegative (thus "nonnegative operator" would be more accurate terminology than "positive operator", but the use of the term "positive operator" has a long history).