Q:

Similarity between special matrices and special complex numbers

From Wikipedia:

It is occasionally useful (but sometimes misleading) to think of the
  relationships of different kinds of normal matrices as analogous to
  the relationships between different kinds of complex numbers:

Invertible matrices are analogous to non-zero complex numbers
Unitary matrices are analogous to complex numbers whose absolute
  value is 1
Hermitian matrices are analogous to real numbers
Hermitian positive definite matrices are analogous to positive real
  numbers
Skew Hermitian matrices are analogous to purely imaginary numbers

I also happen to see that the number analogy of each kind of special matrices also agree with their eigenvalues:

eigenvalues of invertible matrices are non-zero complex numbers
eigenvalues of Unitary matrices are complex numbers whose absolute value is 1
eigenvalues of Hermitian matrices are real numbers
eigenvalues of Hermitian positive definite matrices are positive real numbers
eigenvalues of Skew Hermitian matrices are purely imaginary numbers

I wonder 

if the relation between the eigenvalues of special matrices and
their number analogy is just coincidence, or there are something
inherent, fundamental and more than analogy?
if there are other ways than eigenvalues by which the special
matrices and special numbers are similar to each other? (I actually
don't quite understand what Wikipedia means by analogy. The example
in terms of eigenvalues is just my guess. The author may have other
things in mind and there may be other possibilities.)
if there are other kinds of special matrices not mentioned in the
list have similar analogy to special numbers? (Hermitian negative definite is too trivial to mention.)
if there are some relevant references? (It seems that this kind of analogy is mentioned in
Halmos "Linear Algebra", but cannot find where it is.)

Thanks and regards! 

A:

The complex numbers and the algebra of all $n \times n$ complex matrices are particular examples of (complex) Banach algebras with involution. The involution is somewhat (though not always perfectly) analogous to complex conjugation. The algebra of bounded linear operators on a Hilbert space is one example of such an algebra (mentioned in the reference of Halmos in Theo's comment), which includes both these cases as subcase. In that case, the involution sends an operator to its adjoint, and the analogy with complex conjugation is quite strong. The behaviour of an operator with respect to this involution is strongly reflected in the behaviour of its spectrum (in the case of finite dimensional vector spaces, spectrum of a matrix equates to the set of eigenvalues of that matrix).
However, the complex numbers is unique among (complex) Banach algebras, in that the only Banach division algebra (that is, Banach algebra in which every non-zero element has
a multiplicative inverse) up to isomorphism is the complex numbers itself (this is the Banach-Mazur theorem). 

