do the complex matrices with non zero real eigenvalues form a group

Question

Do the complex matrices with non zero real eigenvalues form a group? Put another way, say $A$ and $B$ are complex $n\times n$ matrices with non zero real eigenvalues, does this imply that $A*B$ also has non zero real eigenvalues?

Answer 1

No, this is not a group with respect to the operation of matrix multiplication because it isn't closed (unless $n=1$, in which case this group is $\mathbb{R}^\times$). Take for example:

$$A=\begin{pmatrix}-1&2\\1&-1\end{pmatrix},\ B=\begin{pmatrix} 7&4\\3&-1\end{pmatrix}$$

$A$ and $B$ have exclusively non-zero real eigenvalues but $AB$ does not.

Answer 2

As pointed out in the other answer, the product of two (real or complex) matrices with real spectra does not necessarily possess a real spectrum. A simple example is given by $$ \pmatrix{-1&0\\ 0&1}\pmatrix{0&1\\ 1&0}=\pmatrix{0&-1\\ 1&0}. $$ Since every real square matrix is a product of two real symmetric matrices (cf. sec. 2 of Olga Taussky (1968), Positive-Definite Matrices and Their Role in the Study of the Characteristic Roots of General Matrices, Advances in Mathematics in Mathematics, 2(2):175-186), the answer to your question is negative for all $n\ge2$.