Name of matrix whose eigenvalues are all conjugate pairs

Question

I've read that "A matrix is positive definite if it’s symmetric and all its eigenvalues are positive"

As the title implies...is there a name for a square matrix whose eigenvalues are all conjugate pairs?

Answer 1

In general, there is no such name for a matrix whose eigenvalues come in conjugate pairs. I think this is in part due to the fact that the typical scenario for this to occur is rather specific, namely the following: if an $n \times n$ real matrix $A$ has complex eigenvalues, they will always occur in complex conjugate pairs.

Now obviously, there are many real matrices that have no complex eigenvalues (i.e., matrices whose eigenvalues are all purely real). Further, there are plenty of complex matrices whose eigenvalues do not come in conjugate pairs, a straightforward example being $$\left(\begin{array}{cc} i & 0 \\ 0 & 2i \end{array} \right).$$ As an interesting aside, there are complex matrices whose eigenvalues are purely real, for example Hermitian matrices, $n \times n$ complex matrices $A$ such that $A = A^*$, where $A^*$ denotes the conjugate transpose of $A$. This property of having purely real eigenvalues is shared by real symmetric matrices.

So the class of matrices whose eigenvalues always come in conjugate pairs is restricted to matrices with complex eigenvalues. And in general there isn't a name for such matrices. Hope that helps.