I recently asked "What is the Metric Tensor?" and a very helpful answer from @R.N.Raia gave me a much better understanding as to what it is. The only problem is that there are a few terms their definition that I don't completely understand. One of which being the Minkowski Metric. Defined by the matrix: $$ \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} = \eta _{\upsilon \nu } $$
I messed around with it and did some multiplication and all those things, but I don't completely understand all of its applications and properties. I also don't really understand how it works not as a matrix. E.g.: the Kronecker Delta is defined by the identity matrix, but it isn't really applied as an identity matrix and more of a term to make sure you don't get unwanted terms in a summation equation like this one: $$ \delta _{m n} \sum_{mn} dx^{m} dx^{n} = ds^{2} $$ So, if y'all could give me an example like that one it will be greatly appreciated.
If you don't know how it works as a matrix, it's just a matrix associated to a bilinear form. Given a vector $v$ you calculate $v^{T}Mv$ with the usual row per column product, $M$ is the matrix you have written above, you can easy check that the result of this calculation is a number. If you want the expression for all possible vectors you get $x^2+y^2+z^2-t^2$ as a quadratic form.