Some properties of a QQ-vector space given the existence of a linear transformation with given properties

Let V be a finite dimensional vector space over \mathbb{Q} and suppose T is a nonsingular linear transformation on V such that T^{-1} = T^2 + T. Prove that the dimension of V is divisible by 3. If the dimension of V is precisely 3, prove that all such transformations are similar.


If T is such a transformation, then we have 1 = T^3 + T^2, and so T^3 + T^2 - 1 = 0. So the minimal polynomial of T divides p(x) = x^3 + x^2 - 1. Now p(x) is irreducible over \mathbb{Q} by the Rational Root Test (Prop. 11 on page 308 in D&F). So the minimal polynomial of T is precisely p(x). Now the characteristic polynomial of T divides some power of p(x) (Prop. 20 in D&F) and so has degree 3k for some k. But of course the degree of the characteristic polynomial is precisely the dimension of V, as desired.

Now if V has dimension 3, then the minimal polynomial of V is the characteristic polynomial, so that the invariant factors of T are simply p(x). In particular, all such T are similar.

Advertisements
Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: