The outer product

The outer product is interesting because it is again the matrix multiplication of two vectors, but this time, the vectors are reversed, producing a matrix. The inner product uses the matrix multiplication of two vectors to get a scalar. The outer product uses two vectors to produce a matrix. Formally, the outer product is defined to be:

If you remember matrix multiplication from Chapter 2, The Matrix, we have the following situation when multiplying an m × n matrix and an n × p matrix. They produce an m × p matrix, as shown in the following diagram. Since we are dealing with vectors, we have an m × 1 matrix and a 1 × p matrix:

Figure 8.2 – The schematics of matrix multiplication

Let's look at an example. First, we have two vectors |u⟩ and |v⟩:

Now, let's do the outer product with them:

Is the outer product...