homework and exercises - Index notation matrix calculation (Intro to Relativity)

Consider the next matrix:

$$M_{ab} = \left(\begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$$ and $$N_{ab} = \left(\begin{array}{cccc} 0 & 0 & \frac{-1}{2} & 0 \\ 0 & 1 & 0 & 0 \\ \frac{1}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}\right)$$

Calculate: $ I_M = M_{ab}\Delta x^a\Delta x^b $ and $ I_N = N_{ab}\Delta x^a\Delta x^b $ for the next subtractions:

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  $a) \Delta x^a = (1,0,1,0) $ and

  
  


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  $ b) \Delta x^a = (1,0,0,0) $

  
  

The problem is I don't really understand the notation nor, given $M_{ab}$ and $N_{ab}$ (as well as $\Delta x^a$), how to operate the matrix.

Any help on this particular example or a general procedure to operate this will be really appreciated.

Answer

Let's look at $M$ and $N$ first. Here, $a$ refers to the row, and $b$ to the column. So, $M_{11}$ corresponds to the upper left corner, and has a value of $-1$. $N_{13}$ corresponds to row $1$, column $3$, with a value of $-\frac{1}{2}$.

Now, lets look at $\Delta x$. This is very similar, but there's only one dimension to work with. So, for b), $\Delta x^{1} = 1$, while for $a = 2, 3, 4$ the value is $0$.

Finally, let's put everything together. The basic concept is that, when an index occurs in both the top and bottom of an equation, there is an implied summation over all values of that index. See below:

$ X_{\gamma} = \begin{bmatrix}1 \\ 2 \\3 \end{bmatrix}$, $Y^{\gamma} = \begin{bmatrix}4 & 5 & 6 \end{bmatrix}$

Then,

$$X_{\gamma}Y^{\gamma} = \sum_{\gamma = 1}^{3} X_{\gamma}Y^{\gamma} = X_{1}Y^{1} + X_{2}Y^{2} + X_{3}Y^{3} = (1*4) + (2*5) + (3*6)$$

Your problem is a bit more complex, as it includes a double summation.

$$M_{ab}x^{a}x^{b} = \sum_{a = 1}^{4} \sum_{b = 1}^{4} M_{ab}x^{a}x^{b}$$

So, now you're out of Einstein notation and in the somewhat more familiar realm of double summations. In case you need a refresher, first let $a=1$, and let $b$ cycle through $1 \to 4$. Now let $a = 2$, and repeat. Add up all these terms, and you have your answer. Alternatively, there are plenty examples of double summations online, the "Einstein notation" part is just understanding how to write them as such.