# Scalar product and isospace

#### parton

In some textbooks you can find that a term
$$\vec{\tau} \cdot \vec{A}_{\mu} = \sum_{a=1}^{3} \tau_{a} \, A_{\mu}^{a}$$
is called scalar product in isospace (where the tau's denotes the Pauli matrices and $$A_{\mu}^{a}$$ 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: $$\gamma^{\mu} \, A_{\mu}$$ 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?

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