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quadratic equations on 2 by 2 matrices
Differences between Real Matrices and Complex matrices.Systems of equations with matricesHow to determine if a set is a subspace of the vector space of all complex $2times 2$ matrices?Counting Distinct matricesOn multiplying symmetric matrices by diagonal matrices with roots of unityFind Matrix C relating two symmetric matrices A and B (Such that AC = B)Structure-preserving matricesIs there a case that 2 positive matrices multiplication returning 0 element?
.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty
margin-bottom:0;
.everyonelovesstackoverflowposition:absolute;height:1px;width:1px;opacity:0;top:0;left:0;pointer-events:none;
$begingroup$
For real non-zero $2times 2$ matrices, can we say:
For any $A,B$, there are at most two matrices $X$ such that $XX + AX + B =0$
Is there a way to see this without going in the direction of writing the open form result of $XX + AX + B$ and solve each element is equal to zero, etc.?
linear-algebra matrices quadratic-forms
$endgroup$
add a comment
|
$begingroup$
For real non-zero $2times 2$ matrices, can we say:
For any $A,B$, there are at most two matrices $X$ such that $XX + AX + B =0$
Is there a way to see this without going in the direction of writing the open form result of $XX + AX + B$ and solve each element is equal to zero, etc.?
linear-algebra matrices quadratic-forms
$endgroup$
add a comment
|
$begingroup$
For real non-zero $2times 2$ matrices, can we say:
For any $A,B$, there are at most two matrices $X$ such that $XX + AX + B =0$
Is there a way to see this without going in the direction of writing the open form result of $XX + AX + B$ and solve each element is equal to zero, etc.?
linear-algebra matrices quadratic-forms
$endgroup$
For real non-zero $2times 2$ matrices, can we say:
For any $A,B$, there are at most two matrices $X$ such that $XX + AX + B =0$
Is there a way to see this without going in the direction of writing the open form result of $XX + AX + B$ and solve each element is equal to zero, etc.?
linear-algebra matrices quadratic-forms
linear-algebra matrices quadratic-forms
edited 8 hours ago
independentvariable
asked 8 hours ago
independentvariableindependentvariable
3631 silver badge11 bronze badges
3631 silver badge11 bronze badges
add a comment
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add a comment
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2 Answers
2
active
oldest
votes
$begingroup$
No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
$$
pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
$$
for any $t$.
Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
$$
Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
$$
And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.
$endgroup$
$begingroup$
how did you come up with this so fast? may I ask your technique?
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
@independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
$endgroup$
– user1551
7 hours ago
add a comment
|
$begingroup$
No, take for example $A=0$ and
$$
B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
$$
Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.
$endgroup$
$begingroup$
The matrices are non zero though.
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
But the question is about showing it can have more than 2, isnt it?
$endgroup$
– independentvariable
7 hours ago
1
$begingroup$
Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
Still, good for knowledge :)
$endgroup$
– independentvariable
7 hours ago
add a comment
|
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2 Answers
2
active
oldest
votes
2 Answers
2
active
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active
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votes
active
oldest
votes
$begingroup$
No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
$$
pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
$$
for any $t$.
Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
$$
Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
$$
And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.
$endgroup$
$begingroup$
how did you come up with this so fast? may I ask your technique?
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
@independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
$endgroup$
– user1551
7 hours ago
add a comment
|
$begingroup$
No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
$$
pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
$$
for any $t$.
Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
$$
Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
$$
And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.
$endgroup$
$begingroup$
how did you come up with this so fast? may I ask your technique?
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
@independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
$endgroup$
– user1551
7 hours ago
add a comment
|
$begingroup$
No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
$$
pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
$$
for any $t$.
Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
$$
Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
$$
And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.
$endgroup$
No. E.g. over $mathbb R$, we have $pmatrix1&t\ 0&-1^2=I$ for any $t$. It also follows that
$$
pmatrix1&t\ 0&-1^2+pmatrix0&0\ 0&1pmatrix1&t\ 0&-1+pmatrix-1&0\ 0&0=0
$$
for any $t$.
Edit. Another counterexample (inspired by Dietrich Burde's answer). Let $u,v$ be any two vectors such that $v^Tu=0$. Then $X=uv^T$ is a solution to the equation $X^2=0$. In particular, if the underlying field is $GF(2)$ (the field consisting of only two elements $0$ and $1$, with $1+1=0$), the equation $Y^2=0$ for a $2times2$ matrix $Y$ has exactly four solutions
$$
Yinleftpmatrix0&0\ 0&0, pmatrix0&1\ 0&0, pmatrix0&0\ 1&0, pmatrix1&1\ 1&1right.
$$
And so does the equation $X^2=I$ over $GF(2)$: just put $X=Y+I$.
edited 7 hours ago
answered 8 hours ago
user1551user1551
81.1k6 gold badges74 silver badges137 bronze badges
81.1k6 gold badges74 silver badges137 bronze badges
$begingroup$
how did you come up with this so fast? may I ask your technique?
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
@independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
$endgroup$
– user1551
7 hours ago
add a comment
|
$begingroup$
how did you come up with this so fast? may I ask your technique?
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
@independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
$endgroup$
– user1551
7 hours ago
$begingroup$
how did you come up with this so fast? may I ask your technique?
$endgroup$
– independentvariable
8 hours ago
$begingroup$
how did you come up with this so fast? may I ask your technique?
$endgroup$
– independentvariable
8 hours ago
1
1
$begingroup$
@independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
$endgroup$
– user1551
7 hours ago
$begingroup$
@independentvariable It may not be hard for someone to figure out a counterexample, but I just happen to know that $X^2=I$ has infinitely many solutions over a field of characteristic zero.
$endgroup$
– user1551
7 hours ago
add a comment
|
$begingroup$
No, take for example $A=0$ and
$$
B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
$$
Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.
$endgroup$
$begingroup$
The matrices are non zero though.
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
But the question is about showing it can have more than 2, isnt it?
$endgroup$
– independentvariable
7 hours ago
1
$begingroup$
Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
Still, good for knowledge :)
$endgroup$
– independentvariable
7 hours ago
add a comment
|
$begingroup$
No, take for example $A=0$ and
$$
B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
$$
Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.
$endgroup$
$begingroup$
The matrices are non zero though.
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
But the question is about showing it can have more than 2, isnt it?
$endgroup$
– independentvariable
7 hours ago
1
$begingroup$
Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
Still, good for knowledge :)
$endgroup$
– independentvariable
7 hours ago
add a comment
|
$begingroup$
No, take for example $A=0$ and
$$
B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
$$
Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.
$endgroup$
No, take for example $A=0$ and
$$
B=beginpmatrix 0 & 1 cr 0 & 0 endpmatrix.
$$
Then $X^2=-B$ has no solution, since $X$ is nilpotent, so that $X^2=0$ but $Bneq 0$.
answered 8 hours ago
Dietrich BurdeDietrich Burde
87.9k6 gold badges50 silver badges111 bronze badges
87.9k6 gold badges50 silver badges111 bronze badges
$begingroup$
The matrices are non zero though.
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
But the question is about showing it can have more than 2, isnt it?
$endgroup$
– independentvariable
7 hours ago
1
$begingroup$
Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
Still, good for knowledge :)
$endgroup$
– independentvariable
7 hours ago
add a comment
|
$begingroup$
The matrices are non zero though.
$endgroup$
– independentvariable
8 hours ago
1
$begingroup$
It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
But the question is about showing it can have more than 2, isnt it?
$endgroup$
– independentvariable
7 hours ago
1
$begingroup$
Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
Still, good for knowledge :)
$endgroup$
– independentvariable
7 hours ago
$begingroup$
The matrices are non zero though.
$endgroup$
– independentvariable
8 hours ago
$begingroup$
The matrices are non zero though.
$endgroup$
– independentvariable
8 hours ago
1
1
$begingroup$
It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
It doesn't matter really. You can adapt this example with nonzero $A$. Quadratic matrix solutions need not have any solution.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
But the question is about showing it can have more than 2, isnt it?
$endgroup$
– independentvariable
7 hours ago
$begingroup$
But the question is about showing it can have more than 2, isnt it?
$endgroup$
– independentvariable
7 hours ago
1
1
$begingroup$
Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
Yes, sorry, I thought of "at least". So quadratic matrix equations can have no solution, or a number of solutions including infinity. It comes from the fact that the matrix algebra is not commutative.
$endgroup$
– Dietrich Burde
7 hours ago
$begingroup$
Still, good for knowledge :)
$endgroup$
– independentvariable
7 hours ago
$begingroup$
Still, good for knowledge :)
$endgroup$
– independentvariable
7 hours ago
add a comment
|
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