Here's an extension of a list posted earlier. If anybody can think of any additions to the list, please post :D!
Perspectives of the world:
Optimist – The glass is half-full.
Pessimist – The glass is half-empty.
Existentialist – The glass is.
Fatalist – The...
Given two matrices, A and B. Is the following statement true?
A is similar to B \Longleftrightarrow A^k is similar to B^k.
By definition, A and B are similar if there exists an invertible matrix P such that B = P-1AP.
The Attempt at a...
1. Prove that the rank of a matrix is invariant under similarity.
Notes so far:
Let A, B, P be nxn matrices, and let A and B be similar. That is, there exists an invertible matrix P such that B = P-1AP. I know the following relations so far: rank(P)=rank(P-1)=n ; rank(A) = rank(AT); rank(A) +...