Two complex matrices with same rank are similar?

Question

If two matrices with pure nonzero complex entries

in $M_n( \Bbb{C}) $

are of the same rank, then can we say that both matrices are similar? Edit:

Where the term 'pure nonzero complex entries' indicates that $\forall \ a_{i,j}$ entry in $A \in M_n( \Bbb{C} ), a_{i,j} \ $ has nonzero imaginary part. Where $i$ runs from $1,2 \cdots ,n$ , and so is $j$

If you're not happy with the answers below, consider the $1\times 1$ case: the rank one 'matrices' $(i)$ and $(2i)$ are not similar. — peter a g, Dec 11 '19 at 19:26
Your counterexample is kind of trivial but satisfying! Can you explain further, what minimal extra conditions do I need to bridge the gap!? — Alfha, Dec 11 '19 at 19:34

score 2 · Answer 1 · answered Dec 11 '19 at 18:50

2

Hint: What can you say of the determinant of similar matrices?

answered Dec 11 '19 at 18:50

Fimpellizzeri

23,321

Determinant is necessary condition for similarity of two matrices, not sufficient condition! – Alfha Dec 11 '19 at 18:53
Since it's necessary, if you can find two matrices that fit your description but have different determinants, then...? – Fimpellizzeri Dec 11 '19 at 18:54
That is where I'm stuck! How to construct such matrices? – Alfha Dec 11 '19 at 18:56
By chance, it would be very unlikely to stumble upon matrices with the same determinant. – Fimpellizzeri Dec 11 '19 at 18:58
You aren't understanding me! If I'll be able to construct one or two such pair of matrices that doesn't mean I've the power to assume it as generalized result ! – Alfha Dec 11 '19 at 19:06
I think you misunderstand. The generalized result you seek to prove is false; to that end, all that's needed is to provide a counter example. If instead you wish to ask 'under what conditions can we say that matrices with these properties are similar?', then your question should reflect that. – Fimpellizzeri Dec 11 '19 at 19:08
Let us continue this discussion in chat. – Alfha Dec 11 '19 at 19:25

José Carlos Santos · Accepted Answer · 2019-12-11T20:08:44.487

2

No. Take, for instance,$$A=\begin{bmatrix}2i&i\\-2i&-i\end{bmatrix}\text{ and }B=\begin{bmatrix}i&i\\-i&-i\end{bmatrix}.$$Both of them have rank $1$. But they are not similar. For instance, $A^2\neq0$, but $B^2=0$.

edited Dec 11 '19 at 20:08

answered Dec 11 '19 at 18:51

José Carlos Santos

440,053

2

There was something in there about nonzero entries. – GEdgar Dec 11 '19 at 18:53
I missed that and I've edited my answer. Thank you. – José Carlos Santos Dec 11 '19 at 19:06
Pure complex entries means every entry has nonzero imaginary part! – Alfha Dec 11 '19 at 19:08
@Alfha Is it fine now? – José Carlos Santos Dec 11 '19 at 20:09
Yes! And eigen values and trace of both matrices aren't same too! – Alfha Dec 13 '19 at 13:56

Two complex matrices with same rank are similar?

2 Answers2