# 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

1. 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?
2. 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.)
3. 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.)
4. 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!

• Your entire list is based on the identification of $\mathbb{C}$ with the complex $1 \times 1$ matrices. The analogy is obtained by replacing $1$ by $n$ throughout. I'll remove the (elementary-number-theory) tag and before you ask: elementary number theory is generally used for properties of natural numbers or integers, divisibility, etc. – t.b. Aug 21 '11 at 22:56
• Something related... – J. M. is a poor mathematician Aug 21 '11 at 22:56
• Re: reference in Halmos: section 71, page 139 "[...] Hermitian matrices play the same role as real numbers." It seems that much of the part on orthogonality is built around the analogy you ask about. – t.b. Aug 21 '11 at 23:01
• @Theo: Thanks! The first quote mentioned "different kinds of normal matrices as analogous to the relationships between different kinds of complex numbers" followed by "invertible matrices are analogous to non-zero complex numbers". I was wondering if invertible matrices are not all normal matrices, and the quote meant "invertible normal matrices" when saying "invertible matrices"? – Tim Aug 23 '11 at 22:26
• Normal matrices are diagonalizable, invertible matrices aren't in general: e.g. $\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}$ is invertible but not normal (check this!). No, I don't think Halmos meant to write invertible normal matrices, as the invertible normal $n \times n$-matrices don't form a group under matrix multiplication if $n \geq 2$ while the invertible matrices do. – t.b. Aug 23 '11 at 22:31

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).
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.