The discussion revolves around the relationship between the eigenvalues of a matrix A and its inverse A^(-1), particularly when A is similar to A^(-1). The original poster questions whether this similarity implies that all eigenvalues must be 1 or -1.
The discussion is active, with participants providing insights and questioning assumptions. Some have offered guidance on the nature of eigenvalues in relation to similarity, while others are reflecting on the implications of their reasoning. There is no explicit consensus, but multiple interpretations and approaches are being explored.
Participants note that the assumption of eigenvalues being 1 or -1 may not hold for all matrices, particularly larger ones, and that diagonal matrices could serve as counterexamples. The original poster's confusion about the textbook's assertion is acknowledged, indicating a need for deeper exploration of the topic.
jbunniii said:Yes, the eigenvalues will be 1 or -1.
First, if A and B are similar matrices, what can you say about their eigenvalues?
Second, if \lambda is an eigenvalue of A, what number must be an eigenvalue of A^(-1)?
Dick said:Suppose the eigenvalues are say, 2 and 1/2?
Sanglee said:Oh~~~~~~~~I got it! it's really simple question, but I thought it was complicated. hehe
So, A and inverse of A are similar, so their eigenvalues are same.
if one of A's eigenvalues is n, a eigenvalues of its inverse will be 1/n.
But the two matrices are similar, so n=1/n
Then, n^2=1, so n=1or-1
Is it right?
Thanks guys!