The Spectrum of a Linear Operator

Let M be a square matrix with complex elements. σ(M) is the ‘spectrum’ of M which is the set of eigen values of M.
x∈σ(M) ↔ (x is an eigenvalue of M). With the Cayley-Hamilton theorem you can turn these anaolgies in to theorems:

M is singular ↔ 0 ∈ σ(M).
x ∈ σ(M) ↔ conjugate x ∈ σ(conjugate transpose M).
M ∈ unitary ↔ σ(M) ⊂ unit circle of complex plane.
M ∈ Hermitian ↔ σ(M) ⊂ real axis of complex plane.
M ∈ positive definite Hermitian ↔ σ(M) ⊂ positive real axis of complex plane.
M ∈ skew Hermitian ↔ σ(M) ⊂ imaginary axis of complex plane.

An orthogonal matrix is a real unitary matrix.
Unimodular matrices are variously defined.