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.