Set of linear operators problem
0
Let S be a set of linear operators $ in L(mathbb{C^3}) $ so that no two linear operators are similar and so that
$$ A^{10} +3*A^9 +2*A^8 = 0 , forall A in S $$
How many elements can S contain?
I tried by looking at similarity invariants but don't really know what elements of S look like
linear-algebra
asked Nov 28 '18 at 13:28
user15269
1608
add a comment |
0
Let S be a set of linear operators $ in L(mathbb{C^3}) $ so that no two linear operators are similar and so that
$$ A^{10} +3*A^9 +2*A^8 = 0 , forall A in S $$
How many elements can S contain?
I tried by looking at similarity invariants but don't really know what elements of S look like
linear-algebra
asked Nov 28 '18 at 13:28
user15269
1608
Can you infer anything from $A^{10}+3A^9+2A^8=A^8(A+2I)(A+I)=0$?
– Yadati Kiran
Nov 28 '18 at 13:32
Then S would contain three linear operators whose matrixes are the solutions of the equation? The three matrixes are not similar because they have different determinants?
– user15269
Nov 28 '18 at 13:41
Not only three. $A^8=0$ does not only imply $A=0$. $A$ can be nilpotent.
– Yadati Kiran
Nov 28 '18 at 13:42
Also yes for your second question. Similar matrices have same determinants. But the converse is not true.
– Yadati Kiran
Nov 28 '18 at 13:45
Seems to me, that uin this case there are not that many nilpotent operators - en.wikipedia.org/wiki/Nilpotent_matrix
– pepa.dvorak
Nov 30 '18 at 10:46
add a comment |
0
0
0
Let S be a set of linear operators $ in L(mathbb{C^3}) $ so that no two linear operators are similar and so that
$$ A^{10} +3*A^9 +2*A^8 = 0 , forall A in S $$
How many elements can S contain?
I tried by looking at similarity invariants but don't really know what elements of S look like
linear-algebra
asked Nov 28 '18 at 13:28
user15269
1608
Let S be a set of linear operators $ in L(mathbb{C^3}) $ so that no two linear operators are similar and so that
$$ A^{10} +3*A^9 +2*A^8 = 0 , forall A in S $$
How many elements can S contain?
I tried by looking at similarity invariants but don't really know what elements of S look like
linear-algebra
linear-algebra
asked Nov 28 '18 at 13:28
user15269
1608
asked Nov 28 '18 at 13:28
user15269
1608
asked Nov 28 '18 at 13:28
user15269
1608
asked Nov 28 '18 at 13:28
user15269
1608
asked Nov 28 '18 at 13:28
user15269
1608
1608
Can you infer anything from $A^{10}+3A^9+2A^8=A^8(A+2I)(A+I)=0$?
– Yadati Kiran
Nov 28 '18 at 13:32
Then S would contain three linear operators whose matrixes are the solutions of the equation? The three matrixes are not similar because they have different determinants?
– user15269
Nov 28 '18 at 13:41
Not only three. $A^8=0$ does not only imply $A=0$. $A$ can be nilpotent.
– Yadati Kiran
Nov 28 '18 at 13:42
Also yes for your second question. Similar matrices have same determinants. But the converse is not true.
– Yadati Kiran
Nov 28 '18 at 13:45
Seems to me, that uin this case there are not that many nilpotent operators - en.wikipedia.org/wiki/Nilpotent_matrix
– pepa.dvorak
Nov 30 '18 at 10:46
add a comment |
Can you infer anything from $A^{10}+3A^9+2A^8=A^8(A+2I)(A+I)=0$?
– Yadati Kiran
Nov 28 '18 at 13:32
Then S would contain three linear operators whose matrixes are the solutions of the equation? The three matrixes are not similar because they have different determinants?
– user15269
Nov 28 '18 at 13:41
Not only three. $A^8=0$ does not only imply $A=0$. $A$ can be nilpotent.
– Yadati Kiran
Nov 28 '18 at 13:42
Also yes for your second question. Similar matrices have same determinants. But the converse is not true.
– Yadati Kiran
Nov 28 '18 at 13:45
Seems to me, that uin this case there are not that many nilpotent operators - en.wikipedia.org/wiki/Nilpotent_matrix
– pepa.dvorak
Nov 30 '18 at 10:46
Can you infer anything from $A^{10}+3A^9+2A^8=A^8(A+2I)(A+I)=0$?
– Yadati Kiran
Nov 28 '18 at 13:32
Can you infer anything from $A^{10}+3A^9+2A^8=A^8(A+2I)(A+I)=0$?
– Yadati Kiran
Nov 28 '18 at 13:32
Then S would contain three linear operators whose matrixes are the solutions of the equation? The three matrixes are not similar because they have different determinants?
– user15269
Nov 28 '18 at 13:41
Then S would contain three linear operators whose matrixes are the solutions of the equation? The three matrixes are not similar because they have different determinants?
– user15269
Nov 28 '18 at 13:41
Not only three. $A^8=0$ does not only imply $A=0$. $A$ can be nilpotent.
– Yadati Kiran
Nov 28 '18 at 13:42
Not only three. $A^8=0$ does not only imply $A=0$. $A$ can be nilpotent.
– Yadati Kiran
Nov 28 '18 at 13:42
Also yes for your second question. Similar matrices have same determinants. But the converse is not true.
– Yadati Kiran
Nov 28 '18 at 13:45
Also yes for your second question. Similar matrices have same determinants. But the converse is not true.
– Yadati Kiran
Nov 28 '18 at 13:45
Seems to me, that uin this case there are not that many nilpotent operators - en.wikipedia.org/wiki/Nilpotent_matrix
– pepa.dvorak
Nov 30 '18 at 10:46
Seems to me, that uin this case there are not that many nilpotent operators - en.wikipedia.org/wiki/Nilpotent_matrix
– pepa.dvorak
Nov 30 '18 at 10:46
add a comment |
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Can you infer anything from $A^{10}+3A^9+2A^8=A^8(A+2I)(A+I)=0$?
– Yadati Kiran
Nov 28 '18 at 13:32
Then S would contain three linear operators whose matrixes are the solutions of the equation? The three matrixes are not similar because they have different determinants?
– user15269
Nov 28 '18 at 13:41
Not only three. $A^8=0$ does not only imply $A=0$. $A$ can be nilpotent.
– Yadati Kiran
Nov 28 '18 at 13:42
Also yes for your second question. Similar matrices have same determinants. But the converse is not true.
– Yadati Kiran
Nov 28 '18 at 13:45
Seems to me, that uin this case there are not that many nilpotent operators - en.wikipedia.org/wiki/Nilpotent_matrix
– pepa.dvorak
Nov 30 '18 at 10:46