Minkowski Inner Product and General Tensor/Matrix Question

[Vorde]

Hello all.

I have a fairly rudimentary knowledge of matrices and broader linear algebra. This gets me in a lot of trouble when I'm following along the math of something fine and then I run into some matrix stuff and get stumped, like this. I'm a little bit confused on taking the inner product from the Minkowski Tensor to the actual number. I understand why (in the context of spacetime intervals) it makes sense to define the metric as diag(-1,1,1,1) = $\eta$_{$\alpha\beta$}

What I don't get is that if you define the inner product of two vectors $A$ and $B$ as $\eta$_{$\alpha\beta$}A^$\alpha$B^$\beta$ (I hope I got the summation convention right), how do you get from the matrix form to the number -t^$\alpha$t^$\beta$+x^$\alpha$x^$\beta$ ...(and so on)?
It is just a rule of matrices I don't know? Or is it a specific thing in this context.

Thank you
Also, unrelated: my textbook didn't explicitly say that the spacetime interval (squared) is equal to the inner product, is that true?

 

[HallsofIvy]

It is, in fact, the standard matrix multiplication that you do know, but the order of multiplication is different- the only order in which those matrices can be multiplied. This would be interpreted as
$$\begin{pmatrix}t & x & y & z\end{pmatrix}\begin{pmatrix}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}t \\ x\\ y \\ z\end{pmatrix}$$
$$= \begin{pmatrix}t & x & y & z\end{pmatrix}\begin{pmatrix}-t \\ x \\ y \\ z\end{pmatrix}= -t^2+ x^2+ y^2+ z^2$$

 

[Vorde]

Ah, you're totally right. I wasn't thinking of the vectors as matrices.

Thank you