Suffix notation

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Suffix notation is a method of notation which is of use when dealing with tensors. Particular examples of tensors include vectors and matrices, and suffix notation can greatly simplify algebraic manipulations involving these types of mathematical object.

The components of a vector (with respect to some co-ordinate system) might be written \boldsymbol{x}=(x_1,x_2,x_3). More concisely, we could write xi for the components of the vector, where i = 1,2,3. To motivate this notation, we will consider the equation Ax = b for some matrix A and vectors x,b. We will use the convention that if A is a matrix, then (A)ij = aij is the element of that matrix in the ith row and jth column. One way of thinking of vector equations is as a shorthand for a set of simultaneous equations - each component of the vectors gives an equation. Explicitly, consider the set of three equations for the three unknowns x1,x2,x3:

a11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3


This can be rewritten in a matrix/vector form as equation Ax = b:


\begin{pmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33} 
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2 \\
x_3 
\end{pmatrix}
=
\begin{pmatrix}
b_1 \\
b_2 \\
b_3 
\end{pmatrix}

Comparison of these two forms should convince you that the "go along the column and down the rows" rule for multiplying a matrix and a vector is sensible. We can also write the above system of equations more succinctly in suffix notation. We notice that in any of the three equations, the first index on the aij elements is fixed whilst the second varies from 1 to 3. Thus:


\sum_{j=1}^3 a_{1j}x_j  = b_1


\sum_{j=1}^3 a_{2j}x_j  = b_2


\sum_{j=1}^3 a_{3j}x_j  = b_3

Even more succinctly, we can write this as the single expression


\sum_{j=1}^3 a_{ij}x_j=b_i

When you see such an equation, remember that it is a shorthand notation for writing three equations at once, for i = 1,2,3 (in 3D).

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