Recently, I found an important matrix in analog circuit domain and it need to be proved diagonalized. Then I try to resolve it into a small problem that is:
if there are a $n\times n$ real symmetric matrix $A$ with rank $m$ ($m<n$) and a $n\times n$ diagonal matrix $B$ with rank $n$, can matrix $AB$ be diagonalized?
I can not solve this problem, I would appreciate it if you can help me. You can ask me for more details about the original problem. Thank you very much!
What a strange question (what is the point of insisting that $A$ not have full rank?). Anyway try $$ A=\pmatrix{0&1&0\cr1&0&0\cr0&0&0\cr},\qquad B=\pmatrix{1&0&0\cr0&-1&0\cr0&0&\sqrt{42}\cr}. $$