scalar product and isospace

parton

In some textbooks you can find that a term
[tex] \vec{\tau} \cdot \vec{A}_{\mu} = \sum_{a=1}^{3} \tau_{a} \, A_{\mu}^{a} [/tex]
is called scalar product in isospace (where the tau's denotes the Pauli matrices and [tex]A_{\mu}^{a}[/tex] is a four-vector). But how can one call this "scalar" product. The product is a matrix and not a scalar. And the usual definition of a scalar product requires that the product has to be a scalar.

Or take another example: [tex] \gamma^{\mu} \, A_{\mu} [/tex] is called a scalar product of four-vectors in space-time. It is confusing. Could anyone explain that to me? Do we really have a scalar product (in a strict mathematical sense) or is it just a convention done by physicists?