Mixing
$(AB) ^3=O_n$ and$(BA) ^3 neq O_n$
$begingroup$
Let $A, B in M_n(mathbb C) $ so that $(AB) ^3=O_n$. Does this imply $(BA)^3=O_n$?.
I saw that the implication is true if $nle 3$ and not true for $nge 4$. What I want is either a counterexample for $nge 4$ or some proof.( I can't find neither of them, I have tried using both HC and taking random matrices)
linear-algebra matrices
asked Jan 28 at 15:41
JustAnAmateurJustAnAmateur
1096
$endgroup$
add a comment |
$begingroup$
Let $A, B in M_n(mathbb C) $ so that $(AB) ^3=O_n$. Does this imply $(BA)^3=O_n$?.
I saw that the implication is true if $nle 3$ and not true for $nge 4$. What I want is either a counterexample for $nge 4$ or some proof.( I can't find neither of them, I have tried using both HC and taking random matrices)
linear-algebra matrices
asked Jan 28 at 15:41
JustAnAmateurJustAnAmateur
1096
$endgroup$
$begingroup$
By $O_n$, do you mean the $ntimes n$ zero matrix?
$endgroup$
– Arthur
Jan 28 at 15:43
$begingroup$
Yes, that is what I mean by it.
$endgroup$
– JustAnAmateur
Jan 28 at 15:47
add a comment |
$begingroup$
Let $A, B in M_n(mathbb C) $ so that $(AB) ^3=O_n$. Does this imply $(BA)^3=O_n$?.
I saw that the implication is true if $nle 3$ and not true for $nge 4$. What I want is either a counterexample for $nge 4$ or some proof.( I can't find neither of them, I have tried using both HC and taking random matrices)
linear-algebra matrices
asked Jan 28 at 15:41
JustAnAmateurJustAnAmateur
1096
$endgroup$
Let $A, B in M_n(mathbb C) $ so that $(AB) ^3=O_n$. Does this imply $(BA)^3=O_n$?.
I saw that the implication is true if $nle 3$ and not true for $nge 4$. What I want is either a counterexample for $nge 4$ or some proof.( I can't find neither of them, I have tried using both HC and taking random matrices)
linear-algebra matrices
linear-algebra matrices
asked Jan 28 at 15:41
JustAnAmateurJustAnAmateur
1096
asked Jan 28 at 15:41
JustAnAmateurJustAnAmateur
1096
asked Jan 28 at 15:41
JustAnAmateurJustAnAmateur
1096
asked Jan 28 at 15:41
JustAnAmateurJustAnAmateur
1096
asked Jan 28 at 15:41
JustAnAmateurJustAnAmateur
1096
1096
$begingroup$
By $O_n$, do you mean the $ntimes n$ zero matrix?
$endgroup$
– Arthur
Jan 28 at 15:43
$begingroup$
Yes, that is what I mean by it.
$endgroup$
– JustAnAmateur
Jan 28 at 15:47
add a comment |
$begingroup$
By $O_n$, do you mean the $ntimes n$ zero matrix?
$endgroup$
– Arthur
Jan 28 at 15:43
$begingroup$
Yes, that is what I mean by it.
$endgroup$
– JustAnAmateur
Jan 28 at 15:47
$begingroup$
By $O_n$, do you mean the $ntimes n$ zero matrix?
$endgroup$
– Arthur
Jan 28 at 15:43
$begingroup$
By $O_n$, do you mean the $ntimes n$ zero matrix?
$endgroup$
– Arthur
Jan 28 at 15:43
$begingroup$
Yes, that is what I mean by it.
$endgroup$
– JustAnAmateur
Jan 28 at 15:47
$begingroup$
Yes, that is what I mean by it.
$endgroup$
– JustAnAmateur
Jan 28 at 15:47
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
One classical counterexample is given by
$$
A=pmatrix{0&1\ &0&1\ &&0&1\ &&&0},
B=pmatrix{1\ &1\ &&1\ &&&0},
$$
so that $AB=J_3(0)oplus0$ and $BA=A=J_4(0)$, where $J_k(0)$ denotes the $ktimes k$ nilpotent Jordan block.
answered Jan 28 at 15:54
user1551user1551
73.9k566129
$endgroup$
$begingroup$
How did you come up with them?
$endgroup$
– JustAnAmateur
Jan 28 at 16:00
2
$begingroup$
@JustAnAmateur It's just a generalisation of the classical example of a pair of $(A,B)$ such that $AB=0$ but $BAne0$.
$endgroup$
– user1551
Jan 28 at 16:02
$begingroup$
Thank you for your insight!
$endgroup$
– JustAnAmateur
Jan 28 at 16:03
add a comment |
Your Answer
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
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votes
$begingroup$
One classical counterexample is given by
$$
A=pmatrix{0&1\ &0&1\ &&0&1\ &&&0},
B=pmatrix{1\ &1\ &&1\ &&&0},
$$
so that $AB=J_3(0)oplus0$ and $BA=A=J_4(0)$, where $J_k(0)$ denotes the $ktimes k$ nilpotent Jordan block.
answered Jan 28 at 15:54
user1551user1551
73.9k566129
$endgroup$
$begingroup$
How did you come up with them?
$endgroup$
– JustAnAmateur
Jan 28 at 16:00
2
$begingroup$
@JustAnAmateur It's just a generalisation of the classical example of a pair of $(A,B)$ such that $AB=0$ but $BAne0$.
$endgroup$
– user1551
Jan 28 at 16:02
$begingroup$
Thank you for your insight!
$endgroup$
– JustAnAmateur
Jan 28 at 16:03
add a comment |
$begingroup$
One classical counterexample is given by
$$
A=pmatrix{0&1\ &0&1\ &&0&1\ &&&0},
B=pmatrix{1\ &1\ &&1\ &&&0},
$$
so that $AB=J_3(0)oplus0$ and $BA=A=J_4(0)$, where $J_k(0)$ denotes the $ktimes k$ nilpotent Jordan block.
answered Jan 28 at 15:54
user1551user1551
73.9k566129
$endgroup$
$begingroup$
How did you come up with them?
$endgroup$
– JustAnAmateur
Jan 28 at 16:00
2
$begingroup$
@JustAnAmateur It's just a generalisation of the classical example of a pair of $(A,B)$ such that $AB=0$ but $BAne0$.
$endgroup$
– user1551
Jan 28 at 16:02
$begingroup$
Thank you for your insight!
$endgroup$
– JustAnAmateur
Jan 28 at 16:03
add a comment |
$begingroup$
One classical counterexample is given by
$$
A=pmatrix{0&1\ &0&1\ &&0&1\ &&&0},
B=pmatrix{1\ &1\ &&1\ &&&0},
$$
so that $AB=J_3(0)oplus0$ and $BA=A=J_4(0)$, where $J_k(0)$ denotes the $ktimes k$ nilpotent Jordan block.
answered Jan 28 at 15:54
user1551user1551
73.9k566129
$endgroup$
One classical counterexample is given by
$$
A=pmatrix{0&1\ &0&1\ &&0&1\ &&&0},
B=pmatrix{1\ &1\ &&1\ &&&0},
$$
so that $AB=J_3(0)oplus0$ and $BA=A=J_4(0)$, where $J_k(0)$ denotes the $ktimes k$ nilpotent Jordan block.
answered Jan 28 at 15:54
user1551user1551
73.9k566129
answered Jan 28 at 15:54
user1551user1551
73.9k566129
answered Jan 28 at 15:54
user1551user1551
73.9k566129
answered Jan 28 at 15:54
user1551user1551
73.9k566129
73.9k566129
$begingroup$
How did you come up with them?
$endgroup$
– JustAnAmateur
Jan 28 at 16:00
2
$begingroup$
@JustAnAmateur It's just a generalisation of the classical example of a pair of $(A,B)$ such that $AB=0$ but $BAne0$.
$endgroup$
– user1551
Jan 28 at 16:02
$begingroup$
Thank you for your insight!
$endgroup$
– JustAnAmateur
Jan 28 at 16:03
add a comment |
$begingroup$
How did you come up with them?
$endgroup$
– JustAnAmateur
Jan 28 at 16:00
2
$begingroup$
@JustAnAmateur It's just a generalisation of the classical example of a pair of $(A,B)$ such that $AB=0$ but $BAne0$.
$endgroup$
– user1551
Jan 28 at 16:02
$begingroup$
Thank you for your insight!
$endgroup$
– JustAnAmateur
Jan 28 at 16:03
$begingroup$
How did you come up with them?
$endgroup$
– JustAnAmateur
Jan 28 at 16:00
$begingroup$
How did you come up with them?
$endgroup$
– JustAnAmateur
Jan 28 at 16:00
2
2
$begingroup$
@JustAnAmateur It's just a generalisation of the classical example of a pair of $(A,B)$ such that $AB=0$ but $BAne0$.
$endgroup$
– user1551
Jan 28 at 16:02
$begingroup$
@JustAnAmateur It's just a generalisation of the classical example of a pair of $(A,B)$ such that $AB=0$ but $BAne0$.
$endgroup$
– user1551
Jan 28 at 16:02
$begingroup$
Thank you for your insight!
$endgroup$
– JustAnAmateur
Jan 28 at 16:03
$begingroup$
Thank you for your insight!
$endgroup$
– JustAnAmateur
Jan 28 at 16:03
add a comment |
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$begingroup$
By $O_n$, do you mean the $ntimes n$ zero matrix?
$endgroup$
– Arthur
Jan 28 at 15:43
$begingroup$
Yes, that is what I mean by it.
$endgroup$
– JustAnAmateur
Jan 28 at 15:47