[Math] Matrix for the reflection over the null space of a matrix

linear algebramatricesreflection

I'm working on my latest linear algebra assignment and one question is as follows:

In $\mathbb R^3$ let R be the reflection over the null space of the matrix

A = [4 4 5]

Find the matrix which represents R using standard coordinates.

I am familiar with the fact that the matrix for the orthogonal projection onto a subspace is given by [P] = A($A^TA$)$A^T$, where A is a matrix with columns that form a basis for the space in question. However, I am not familiar with the concept of a "reflection over the null space". How does this compare with the matrix for the orthogonal projection onto the null space? Is it a similar process?
Thanks for any help.

Best Answer

First of all, the formula should be $$P = B(B^TB)^{-1}B^T$$ where the columns of $B$ form of a basis of $ker(A)$.

Think geometrically when solving it. Points are to be reflected in a plane which is the kernel of $A$ (see third item):

  • find a basis $v_1, v_2$ in $ker(A)$ and set up $B = (v_1 \, v_2)$
  • build the projector $P$ onto $ker(A)$ with above formula
  • geometrically the following happens to a point $x = (x_1 \, x_2 \, x_3)$ while reflecting in the plane $ker(A)$: $x$ is split into two parts - its projection onto the plane and the corresponding orthogonal part of $x$. Then flip the direction of this orthogonal part: $$x = Px + (x - Px) \mapsto Px - (x-Px) \rightarrow x \mapsto Px - (I-P)x = (2P-I)x$$ So, the matrix looked for is $$2P-I$$
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