TITLE: Similarity between special matrices and special complex numbers
QUESTION [5 upvotes]: 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!

REPLY [2 votes]: I think what you're noticing about eigenvalues comes down to this: if $\lambda$ is an eigenvalue of $A$, with corresponding eigenvector $\bf v$, then multiplying $\bf v$ by $A$ is the same as multiplying $\bf v$ by $\lambda$, that is, $A{\bf v}=\lambda{\bf v}$. Also, if $A$ has a dominant eigenvalue $\lambda$ (an eigenvalue exceeding all the others in modulus), then the entries of $A^n$ grow as $\lambda^n$ as $n$ increases, and the length of $A^n{\bf v}$ is (in the limit) proportional to $\lambda^n$ for all $\bf v$ outside of a proper subspace. So in these senses, matrices behave like their eigenvalues.