Nilpotent Operators Linear Algebra
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Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let
$T_1, dots , T_n : V rightarrow V$ be pairwise commuting nilpotent operators on $V$.
(a) Show the composition $T_1 T_2 cdots T_n = 0$.
(b) Does this still hold if we drop the hypothesis that the $T_i$ are pairwise commuting?
What I have tried:
I know that for any $T_i$, $ker({T_i}^n) = ker({T_i}^{n+1}) = cdots$. This implies that ${T_i}^n = 0$ for any $i$. I also know that raising each $T_i$ to a power of itself lessens the dimension of the kernel of such a composition. Any help?
linear-algebra matrices nilpotence
edited Feb 9 at 23:13
darij grinberg
11.2k33167
$endgroup$
add a comment |
$begingroup$
Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let
$T_1, dots , T_n : V rightarrow V$ be pairwise commuting nilpotent operators on $V$.
(a) Show the composition $T_1 T_2 cdots T_n = 0$.
(b) Does this still hold if we drop the hypothesis that the $T_i$ are pairwise commuting?
What I have tried:
I know that for any $T_i$, $ker({T_i}^n) = ker({T_i}^{n+1}) = cdots$. This implies that ${T_i}^n = 0$ for any $i$. I also know that raising each $T_i$ to a power of itself lessens the dimension of the kernel of such a composition. Any help?
linear-algebra matrices nilpotence
edited Feb 9 at 23:13
darij grinberg
11.2k33167
$endgroup$
$begingroup$
The claim of (b) is false, as you can check by setting $n = 2$ and $T_1 = begin{pmatrix} 0 & 1 \ 0 & 0 end{pmatrix}$ and $T_2 = begin{pmatrix} 0 & 0 \ 1 & 0 end{pmatrix}$.
$endgroup$
– darij grinberg
Feb 9 at 23:14
3
$begingroup$
Possible duplicate of Pairwise commuting nilpotent matrices: alternative solution needed
$endgroup$
– darij grinberg
Feb 9 at 23:18
$begingroup$
Part (a) is math.stackexchange.com/questions/880429/… .
$endgroup$
– darij grinberg
Feb 9 at 23:18
add a comment |
$begingroup$
Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let
$T_1, dots , T_n : V rightarrow V$ be pairwise commuting nilpotent operators on $V$.
(a) Show the composition $T_1 T_2 cdots T_n = 0$.
(b) Does this still hold if we drop the hypothesis that the $T_i$ are pairwise commuting?
What I have tried:
I know that for any $T_i$, $ker({T_i}^n) = ker({T_i}^{n+1}) = cdots$. This implies that ${T_i}^n = 0$ for any $i$. I also know that raising each $T_i$ to a power of itself lessens the dimension of the kernel of such a composition. Any help?
linear-algebra matrices nilpotence
edited Feb 9 at 23:13
darij grinberg
11.2k33167
$endgroup$
Let $V$ be an $n$-dimensional vector space over an arbitrary field $K$, and let
$T_1, dots , T_n : V rightarrow V$ be pairwise commuting nilpotent operators on $V$.
(a) Show the composition $T_1 T_2 cdots T_n = 0$.
(b) Does this still hold if we drop the hypothesis that the $T_i$ are pairwise commuting?
What I have tried:
I know that for any $T_i$, $ker({T_i}^n) = ker({T_i}^{n+1}) = cdots$. This implies that ${T_i}^n = 0$ for any $i$. I also know that raising each $T_i$ to a power of itself lessens the dimension of the kernel of such a composition. Any help?
linear-algebra matrices nilpotence
linear-algebra matrices nilpotence
edited Feb 9 at 23:13
darij grinberg
11.2k33167
edited Feb 9 at 23:13
darij grinberg
11.2k33167
edited Feb 9 at 23:13
darij grinberg
11.2k33167
edited Feb 9 at 23:13
darij grinberg
11.2k33167
edited Feb 9 at 23:13
darij grinberg
11.2k33167
11.2k33167
asked Dec 9 '18 at 23:21
user624358
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The claim of (b) is false, as you can check by setting $n = 2$ and $T_1 = begin{pmatrix} 0 & 1 \ 0 & 0 end{pmatrix}$ and $T_2 = begin{pmatrix} 0 & 0 \ 1 & 0 end{pmatrix}$.
$endgroup$
– darij grinberg
Feb 9 at 23:14
3
$begingroup$
Possible duplicate of Pairwise commuting nilpotent matrices: alternative solution needed
$endgroup$
– darij grinberg
Feb 9 at 23:18
$begingroup$
Part (a) is math.stackexchange.com/questions/880429/… .
$endgroup$
– darij grinberg
Feb 9 at 23:18
add a comment |
$begingroup$
The claim of (b) is false, as you can check by setting $n = 2$ and $T_1 = begin{pmatrix} 0 & 1 \ 0 & 0 end{pmatrix}$ and $T_2 = begin{pmatrix} 0 & 0 \ 1 & 0 end{pmatrix}$.
$endgroup$
– darij grinberg
Feb 9 at 23:14
3
$begingroup$
Possible duplicate of Pairwise commuting nilpotent matrices: alternative solution needed
$endgroup$
– darij grinberg
Feb 9 at 23:18
$begingroup$
Part (a) is math.stackexchange.com/questions/880429/… .
$endgroup$
– darij grinberg
Feb 9 at 23:18
$begingroup$
The claim of (b) is false, as you can check by setting $n = 2$ and $T_1 = begin{pmatrix} 0 & 1 \ 0 & 0 end{pmatrix}$ and $T_2 = begin{pmatrix} 0 & 0 \ 1 & 0 end{pmatrix}$.
$endgroup$
– darij grinberg
Feb 9 at 23:14
$begingroup$
The claim of (b) is false, as you can check by setting $n = 2$ and $T_1 = begin{pmatrix} 0 & 1 \ 0 & 0 end{pmatrix}$ and $T_2 = begin{pmatrix} 0 & 0 \ 1 & 0 end{pmatrix}$.
$endgroup$
– darij grinberg
Feb 9 at 23:14
3
3
$begingroup$
Possible duplicate of Pairwise commuting nilpotent matrices: alternative solution needed
$endgroup$
– darij grinberg
Feb 9 at 23:18
$begingroup$
Possible duplicate of Pairwise commuting nilpotent matrices: alternative solution needed
$endgroup$
– darij grinberg
Feb 9 at 23:18
$begingroup$
Part (a) is math.stackexchange.com/questions/880429/… .
$endgroup$
– darij grinberg
Feb 9 at 23:18
$begingroup$
Part (a) is math.stackexchange.com/questions/880429/… .
$endgroup$
– darij grinberg
Feb 9 at 23:18
add a comment |
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$begingroup$
The claim of (b) is false, as you can check by setting $n = 2$ and $T_1 = begin{pmatrix} 0 & 1 \ 0 & 0 end{pmatrix}$ and $T_2 = begin{pmatrix} 0 & 0 \ 1 & 0 end{pmatrix}$.
$endgroup$
– darij grinberg
Feb 9 at 23:14
3
$begingroup$
Possible duplicate of Pairwise commuting nilpotent matrices: alternative solution needed
$endgroup$
– darij grinberg
Feb 9 at 23:18
$begingroup$
Part (a) is math.stackexchange.com/questions/880429/… .
$endgroup$
– darij grinberg
Feb 9 at 23:18