07. Index notation
07. Index notation
Index notation
I can write the entries of an m -by-n matrix A as A=(A11A12A13⋯A1nA21A22A23⋯A2n⋮⋮⋮⋮Am1Am2Am3⋯Amn).
One advantage of writing matrices like this is that it gives a compact formula for operations like matrix multiplication: rather than writing out the full matrix AB , we can just write a formula for the ij th entry (AB)ij . Suppose that A is m -by-n and B is n -by-p .
To get (AB)ij , we need to multiply the i th row of A into the j th column of B , in other words: (AB)ij=(Ai1Ai2⋯Ain)(B1jB2j⋮Bnj)
Associativity of matrix multiplication
To demonstrate how useful index notation is, let's prove that matrix multiplication is associative, that is: (AB)C=A(BC).
For the right-hand side: (A(BC))ij=∑kAik(BC)kj
This looks very similar to the formula for the left-hand side, but the indices k and ℓ have been swapped. That doesn't matter: k and ℓ are dummy indices, so we can just rename them. We'll relabel k as ℓ and ℓ as k : (A(BC))ij=∑ℓ∑kAiℓBℓkCkj.
Index notation is very heavily used in subjects like general relativity. For example, the Riemann curvature tensor Rijkℓ is an object with four indices, some up and some down!