# How can the product of two real linear operators be not real?

I'm puzzled about a statement from Diracs book "The principles of quantum mechanics" (§8, p.28):

As a simple examples of this result, it should be noted that, if $\xi$ and $\eta$ are real, in general $\xi\eta$ is not real. This is an important difference from classical mechanics. However, $\xi\eta + \eta\xi$ is real, and so is $i(\xi\eta - \eta\xi)$. Only when $\xi$ and $\eta$ commute is $\xi\eta$ itself also real.

Here $\xi$ and $\eta$ are linear operators, so (I think) these could be represented as matrices. However how can a product of two real matrices be "not real"? What does Dirac mean when he says "is not real"? Is Dirac maybe talking about eigenvalues? So does Dirac mean, that the product of two matrices with real eigenvalues could have imaginary eigenvalues? And does he want to say, that a real symmetric matrix like $\xi\eta + \eta\xi$ and a purely imaginary antisymmetric matrix like $i(\xi\eta - \eta\xi)$ always have real eigenvalues?

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Flip back a page; Dirac uses real to mean Hermitian when talking about linear operators. So you can see that even if $A$ and $B$ are Hermitian, $AB$ won't be Hermitian unless they commute, whereas those linear combinations will be.