Simultaneously Triangularizable Matrices

Question

If 2 complex matrices share an eigenvector are they simultaneously triangularizable?

I know that commuting matrices have this property and one proof relies on a simple construction. Can this be extended to matrices which have a common eigenvector, or a more general condition?

If matrix size is 2, of course. Beyond that, it is not true in general. — Christophe Boilley, Mar 27 '25 at 17:54

score 2 · Accepted Answer · answered Mar 27 '25 at 18:21

The answer is no.

Let $A$ and $B$ be any two matrices that are not simultaneously triangularizable, such as $$ A = \pmatrix{1 & 0\\0 & 0}, \quad B = \pmatrix{0&1\\1&0}. $$ Then the matrices $$ \pmatrix{1 & 0\\0 & A}, \quad \pmatrix{1 & 0\\0 & B} $$ have a common eigenvector but fail to be simultaneously triangularizable.

Simultaneously Triangularizable Matrices

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