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 eigenvalues of M.
x∊σ(M) ↔ (x is an eigenvalue of M). With the Cayley-Hamilton theorem you can turn these analogies 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.
M∊projection ↔ σ(M) ⊂ {0, 1}.

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