I feel like I just fell off the short bus. I found a problem with a setup like this:
"2X2 matrix defined by: $$U=(a_0+i(\sigma \cdot a))/(a_0-i(\sigma\cdot a)) $$ where $a$ is a three-dimensional vector with real components and $a_0$ is a real number."
This looks like a scalar to me. Dot products are always scalar (to my knowledge) and addition, subtraction, and multiplication of two scalars is scalar... where is the 2X2 matrix coming from? How do I define this matrix?