How to construct a matrix to project 4-dimensional space to a 2-dimensional plane?

Question

How to Construct a matrix to project vectors in 4-dimensional space to a 2-dimensional plane that lies in the space and passes through the origin and the points (1,-1, -1, 1) and (1, -1, 1, 1).

Answer 1

Hey Alex, your question is interesting. There is a general method for computing the projection of one vector onto another vector (n-dim space). The formula for the projection matrix for projecting vector v onto vector a is

P = a*a^T. The projection is then v_a = Pv.

In your problem, you are asking about a plane, which has normal vector n. So, I would recommend a three-stage process.

1. Determine the normal vector n describing your plane. Note that this vector will not be unique (think of how there are an infinite number of vectors orthogonal to a line in 3-space).
2. Compute the projection of your vector v onto n: v_n = P_nv.
3. Compute the projection onto the plane (v_p) using the formula v_p = v-v_n.

Now, we can condense this process!

Note that v_p = v-P_nv = (I-P_n)v. So, the matrix you are after is I-P_n = I-nn^T.

Hope this helps!