Some Comments on Matrices and Isometries

1. Suppose

,

so that

and .

We wish to determine what kind of isometry t is. We may observe that U is a reflection matrix, either by using pattern-matching or computing that det(U) = -1. (Note that U is orthogonal.)

Since t is the composition of a reflection and a translation, we know from the work we did with isometries last quarter that t is either a reflection or a glide reflection. The question is how to tell the difference.

The composition of a reflection and a translation is a reflection precisely when the translation vector is perpendicular to the mirror of reflection, but often it is hard to calculate the equation of the mirror line precisely to check this fact.

However, if t is a reflection, then t^2(x) = x. Thus, it suffices to check what function t^2 is.

Note if U is a reflection matrix, . So:

and t is a reflection, instead of a glide reflection.

However: this is not obvious! You do need to check (using some argument; this is the easiest I know) whether a reflection composed with a translation is a reflection or a glide reflection.

2. Suppose you are given a transformation

,

where U is orthogonal and b is a vector, and you are asked to calculate the inverse function .

The function works, since

(and it is easy to check that s(s^-1(x)) = x as well.)

Now, if U is a reflection matrix, we know that U^-1 = U. If U is a rotation matrix, we know that for some q ,

so

since cosine is an even function and sine is an odd function.