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 Caley-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