1
$\begingroup$

Consider two matrices $A, B\in\mathcal{M}_n(\mathbb{C})$, such that $A, B$ has no common eigenvectors. Is it true that for some nonzero $t\in\mathbb{C}$, matrix $A+tB$ is similar to diagonal matrix $\text{diag}(a_1, a_2, \ldots, a_n)$, where $a_i\not = a_j, \forall i\not= j$?

$\endgroup$
0
4
$\begingroup$

It's false for every $n\ge 3$.

Consider $A_n,B_n$ $n\times n$ square matrices with no common eigenvalue (these exist for $n=0$ and $n\ge 2$). Then $$A=\begin{pmatrix} A_2 & 0 & 0\\ 0 & A_2 & 0\\ 0 & 0 & A_{n-4}\end{pmatrix},\qquad B=\begin{pmatrix} B_2 & 0 & 0\\ 0 & B_2 & 0\\ 0 & 0 & B_{n-4}\end{pmatrix}$$ work for $n=4$ and $n\ge 6$.

It remains to do $n=3,5$. Define $$N=\begin{pmatrix} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 &1& 0\end{pmatrix},\qquad N'=\begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & -1\\ 0 &0& 0\end{pmatrix}$$ Then $N$ and $N'$ have no common eigenvector but the plane spanned by $N,N'$ consists of nilpotent matrices. Then the matrices $$A=\begin{pmatrix} N & 0\\ 0 & A_{n-3}\end{pmatrix},\qquad B=\begin{pmatrix} N' & 0\\ 0 & B_{n-3}\end{pmatrix}$$ work for all $n=3$ and $n\ge 5$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.