• The outer product is a form of matrix operation that most instructors have likely said is not possible. So, tell students to get it out of their brains that the outer product is not doable!
• Write down that the outer product is performed as follows.

So, this operation allows us to move from the bra-ket notation of the projection operator $P_{\alpha}$ and find our matrix representation.

• Convince students that this does produce the correct projection operator by having them Compute Several Outer Products.