Is the following statement is True/false regarding similar matrix

Question

Is the following statement is True/false ?

let $A$ be an $n \times n$ matrix with complex entries . Then A is always similiar to an upper triangular matrix .

My Attempt:

i thinks this satement is false because for example i take
$A =\begin{bmatrix} 1& 0 \\0 & 0 \end{bmatrix}$ and upper triangular matrix .$B =\begin{bmatrix} 2& 0 \\0 & 2 \end{bmatrix}$

Both A and B are not similar

is it correct ??

Any hints/solution will be appreciated

thanks u

Answer 1

The Jordan normal form happens to be an upper triangular matrix to which the given matrix is similar, provided, for instance, its characteristic polynomial splits (which it does over $\mathbb C$).