4

(1) Let $\phi : GL(n,\mathbb{R}) \to \mathbb{R}\setminus\{0\}$ be a group homomorphism. I know that $\phi(A)=\mbox{det}(A)$ and $\phi(A)=1$ are two such examples. But, is there any other example of a group homomorphism?

In general,

(2) Let $M(n, \mathbb{R})$ be the set of $n \times n$ real matrices, and let $\phi : M(n, \mathbb{R}) \to \mathbb{R}$ be a homomorphism (i.e., $\phi(AB)=\phi(A)\phi(B)$ for all $A,B \in M(n, \mathbb{R})$). What is an example of such a mapping other than $\phi(A)=\mbox{det}(A)$, $\phi(A)=1$, and $\phi(A)=0$?

Nicole Abe
  • 41
  • $\phi(A)=\det (A)^n$ or $\phi(A)=|\det(A)|^\alpha$. I think those are all but don't ask me why :-) – Quang Hoang Nov 13 '15 at 11:29
  • 1
    Duplicate of http://math.stackexchange.com/questions/727050/is-the-determinant-the-only-group-homomorphism-from-mathrmgl-n-mathbb-r. – lhf Nov 13 '15 at 11:32
  • @lhf Thanks! But how about the second question? Is it a trivial generalization? – Nicole Abe Nov 13 '15 at 11:54

0 Answers0