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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) = [itex]\eta[/itex]_{[itex]\alpha\beta[/itex]}

What I don't get is that if you define the inner product of two vectors [itex]A[/itex] and [itex]B[/itex] as [itex]\eta[/itex]_{[itex]\alpha\beta[/itex]}A^{[itex]\alpha[/itex]}B^{[itex]\beta[/itex]}(I hope I got the summation convention right), how do you get from the matrix form to the number -t^{[itex]\alpha[/itex]}t^{[itex]\beta[/itex]}+x^{[itex]\alpha[/itex]}x^{[itex]\beta[/itex]}....(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?

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# Minkowski Inner Product and General Tensor/Matrix Question

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