Einstein Notation

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Einstein's summation notation is widely used in the 224 Geometry course, not to mention much of modern mathematics and theoretical physics. It is used to simplify long expressions such as $\sum^n_{i=1}\sum^n_{j=1}\sum^n_{k=1}\sum^n_{l=1}\alpha_{ij}u^ku^l$ . It makes statements and equations easier to read, and often allows us to spot patterns or solutions quicker.

1 Notation

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Notation

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Introduction

We will assume (when it is not stated) that we are working in an n-dimensional vector space $M$ over a field $K$ . We will denote basis vectors of $M$ using a lower index, e.g. $u_1, e_4, x_k$ , etc. If we wish to multiply vectors in $M$ by scalars in $K$ , we will use upper indices e.g. $\alpha^1, c^4, \beta^k$ , etc. Finally, we will assume that the vectors $u_1, u_2, \dots, u_n$ form a basis for $M$ .

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Vectors

As a basic example, we know that every vector in $M$ can be written as a linear combination of basis vectors, i.e. for all $x\in M$ , we have $x = x^1u_1 + x^2u_2 + \cdots + x^nu_n = \sum^n_{i=1}x^iu_i$ , for some unique set of scalars $x^1, x^2, \dots, x^n$ . To implement Einstein's notation, we simply drop the summation symbol, $\sum^n_{i=1}$ , and write $x = x^iu_i$ .

The variable $i$ is called a repeated index. We will use letters such as $i, j, k, l$ , etc. as repeated indices. Where we need $k$ separate indices, for example, we may use $i_1, i_2, \dots, i_k$ . However, we will always assume that the repeated indices are summed from 1 to $n$ .

We can relate this notation back to familiar ideas, e.g. the dot product. In $\mathbb{R}^n$ , we define the dot product of $a = (a_1, a_2, \dots, a_n)$ and $b = (b_1, b_2, \dots, b_n)$ by $a\cdot b = a_1b_1 + a_2b_2 + \dots + a_nb_b$ . In the new notation, we have $a = a^iu_i$ and $b = b^ju_j$ , and so we simple write $a\cdot b = a^ib^i$ .

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Matrices

We may also write matrices using this summing notation. Before, we needed to write out (square) matrices as follows:

$A= \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \dots & a_{nn} \end{pmatrix}$

This notation can be quite cumbersome at times, so we will 'convert' it to the above notation. We will denote a matrix $A$ by the symbol $[a^i_j]$ , again where the indices are summed from 1 to $n$ . The upper index (i) will denote the rows and thte lower index (j) will denote the columns. This means that we have reduced the notation of writing out the entire matrix to one symbol.

As well as this, we can use the summation notation for matrix operations. For example, if we have matrices $A$ and B in the old notation, we have that the $ij$ coordinates of the product $C = AB$ will be given by $c_{ij} = \sum^n_{k=1}a_{ik}b_{kj}$ where $a_{ik}$ are the elements of $A$ and $b_{kj}$ are the elements of $B$ . Adopting the summation notation, we have that $A = [a^i_k]$ and $B = [b^k_j]$ . Thus, the elements of the product are given by $c^i_j = \sum^n_{k=1}a^i_kb^k_j = a^i_kb^k_j$ .

Note that we did not label the indices at random: the lower index of $A$ has to be the upper index of $B$ . Also, the upper index of $C$ has to be the upper index of $A$ and the lower index of $C$ has to be the lower index of $B$ .

As a further example, given the matrix $A = [a^i_j]$ we consider what matrix is given by $a^i_ja^j_ka^k_l$ . As matrix multiplication is associative, let $b^i_k = a^i_ja^j_k$ . By above, this is simply the representation of the matrix $A^2$ . Thus we now look at $b^i_ka^k_l$ , which is just the product of $A^2$ and $A$ . Thus, $a^i_ja^j_ka^k_l$ is the matrix $A^3$ .

We defined the trace of a matrix $A$ by $\mathrm{tr}(A) = \sum^n_{i=1}a_{ii}$ , i.e. the sum of the main diagonal. Thus, we now have $\mathrm{tr}(A) = a^i_i$ .

Finally, we introduce an important symbol, known as the Kronecker Delta, defined by:

$\delta^i_j =$	$\quad 1 \quad i=j$
	$\quad 0\quad i\neq j$

This symbol is useful for changing the repeated index. For example $\delta^i_j \alpha^j = \alpha^i$ , or even $\delta^i_ju_i = u_j$ . In matrix notation, the Kronecker Delta is simply the identity matrix.

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Dual Space

As defined before the Dual Space of $M$ is the space of all linear maps from $M$ to $K$ .

We said above that u_i formed a basis for $M$ . We'll define a mapping $u^j : M\rightarrow K$ given by $u^j(x) = x^j$ , where $x = x^iu_i$ . It can be easily shown that this mapping is linear for any $1 \leq j \leq n$ , and so $u^j\in M^*$ .

These mappings have the property that $\langle u^i, u_j \rangle = \delta^i_j$ . It can be readily shown that these $n$ coordinate functions form a basis for $M^*$ .

Note the difference in notation: $u_i$ is a vector and $u^i$ is a mapping. Similarly, we associate the scalar $\alpha^i$ with vectors and the scalar $\alpha_i$ with mappings.