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#1 h Mar 26, 2025, 9:49 AM

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The matrix $A = (a_{ij}) \in Mat_p(\mathbb{C})$ is defined by the conditions
$a_{12} = a_{23} = \dots = a_{(p-1)p} = 1$ and $a_{ij} = 0$ for a set of indices $(i,j)$ .
Prove that there do not exist nonzero matrices $B, C \in Mat_p(\mathbb{C})$ satisfying the equation
$(I_p + A)^n = B^n + C^n.$ $\forall$ $n$ is a postive integer.

This post has been edited 2 times. Last edited by hef4875, Mar 26, 2025, 9:51 AM

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#4 h Mar 26, 2025, 1:02 PM

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It's this problem.

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3560 posts

#5 h Mar 27, 2025, 4:18 PM • 1 Y

Y by MS_asdfgzxcvb

Thanks @ Filipjack.
$\textbf{Proposition.}$ If $I+A=B+C$ and $(I+A)^2=B^2+C^2$ , then $B=I+A,C=0$ or $B=0,C=I+A$ .
$\textbf{Proof.}$ $I+A=B+C$ , $(I+A)^2=B^2+C^2=B^2+C^2+BC+CB$ ;
then $BC+CB=0$ . Since $C=I+A-B$ , we deduce that
(*) $2B-2B^2+AB+BA=0$ .
(*) is a Riccati equation; we associate the $2p\times 2p$ pseudo-Hamiltonian
$M=\begin{pmatrix}-A&2I_p\\0_p&2I_p+A\end{pmatrix}\in M_{2p}(\mathbb{C})$ . There is a one to one correspondence between
the set of solutions of (*) and the set of $p$ -dimensional $M$ -invariant subspaces $E$ s.t.
(1) $E\oplus span(e_{p+1},\cdots,e_{2p})=span(e_1,\cdots,e_{2p})$ .
The Jordan form of $M$ -in the new basis $(f_1,\cdots,f_{2p})$ - is $diag(J_p[2],J_p[0])$ , where
$J_p[\lambda]=\lambda I_p+J_p$ and $J_p$ is the nilpotent Jordan block of dimension $p$ .
$E$ is in one of those $p+1$ forms $span(f_1,\cdots f_q,f_{p+1},\cdots,f_{2p-q}),q=0,\cdots,p$ .
Only the following $2$ spaces verify condition (1):
$\ker(M^p)=span(e_1,\cdots,e_p)$ and $\ker((M-2I_p)^p)$ .
$\bullet$ It's complicated to write a proof of the previous line and I don't have any nice proof.
Thus (*) has $2$ solutions. Luckily, we have two obvious solutions: $B=0$ and
$B=I+A$ . $\square$

This post has been edited 3 times. Last edited by loup blanc, Mar 27, 2025, 8:32 PM

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Etkan

#6 h Friday at 6:00 PM

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loup blanc wrote:

I've read some very poorly written posts before, but yours is exceptional.

For God's holy sake! Can you for once be nice to someone? I'm not even asking for two posts, one is enough.

hef4875 wrote:

The matrix $A = (a_{ij}) \in Mat_p(\mathbb{C})$ is defined by the conditions
$a_{12} = a_{23} = \dots = a_{(p-1)p} = 1$ and $a_{ij} = 0$ for a set of indices $(i,j)$ .
Prove that there do not exist nonzero matrices $B, C \in Mat_p(\mathbb{C})$ satisfying the equation
$(I_p + A)^n = B^n + C^n.$ $\forall$ $n$ is a postive integer.

This statement is perfectly understandable the way it is written, and people do not necessarily have access to English classes (nor are native English speakers) since they are kids or something like that.
You better stop being rude, or I'm going to go through all of your posts and report every single one of them in which you mistreat someone else. I have a feeling that admins and mods will not like that attitude.

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#7 h Friday at 7:18 PM • 1 Y

Y by RobertRogo

@ Etkan, I see you're threatening me; I sincerely pity you. In
https://artofproblemsolving.com/community/c7t290f7h3523769_prove_the_statement_about_matrices
you stupidly attacked @ RobertRogo because he wrote about a previous post "this is of course false".
You were lucky because Robert Rogo is a good man. I don't even argue with you.
I think we could do a collection to buy you a policeman's whistle.

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#8 h Yesterday at 1:00 AM

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loup blanc wrote:

I think we could do a collection to buy you a policeman's whistle.

I think we could all be a little nicer, no?

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1551 posts

Etkan

#9 h Yesterday at 1:02 AM

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loup blanc wrote:

@ Etkan, I see you're threatening me; I sincerely pity you. In
https://artofproblemsolving.com/community/c7t290f7h3523769_prove_the_statement_about_matrices
you stupidly attacked @ RobertRogo because he wrote about a previous post "this is of course false".
You were lucky because Robert Rogo is a good man. I don't even argue with you.
I think we could do a collection to buy you a policeman's whistle.

I said I don't recommend to use a certain wording and I even apologised after in case someone felt I had spoken in an aggressive tone, explicitly saying it. Hence I don't see how it was an attack.
The "policeman's whistle" is a (relatively) original answer, but it is far from being true because I always tried to speak nicely to everyone, apologised when I didn't (for example here), and never reported people until now.
Hence I consider #3 and #7 to be unnecessarily aggressive and I will report them. Please, admins/mods, tell me if I'm wrong.

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