Eigenvalues of a matrix with repeating pattern of entries
Clash Royale CLAN TAG #URR8PPP up vote 10 down vote favorite I observed that if $$A = beginbmatrix a & b \ c & d endbmatrix$$ with non-zero eigenvalues $alpha$ and $beta$, then $$beginbmatrix A & A\ A & A endbmatrix$$ has eigenvalues $2 alpha$, $2 beta$, and $0$. Also, $$beginbmatrix A & A & A \ A & A & A \ A & A & A endbmatrix$$ has eigenvalues $3 alpha$, $3 beta$, $0$. Therefore, my conjecture is that for some $r$, $A^[r]$ has eigenvalues $(r+1) alpha$, $(r+1) beta$, $0$. Is it correct? Is there some theorems related to this? How about their eigenvectors? Can you please send me links that can help me with this kind of problem? PS. This is my first time asking here. I am an undergrad math student. Please help me. Thank u so much. linear-algebra matrices eigenvalues-eigenvectors share | cite | improve this question edited Jul 22 at 11:37 asked Jul 22 at 11:21 Diggie Cruz 51 4 6 these