Mixing
Proof that $M$ and $M^{T}$ are similar
$begingroup$
Triangular matrix $M in mathbb R^{n,n}$ and all elements on the diagonal are different. Proof that $M$ and $M^{T}$ are similar.
I know that the matrices are similar when the matrix similarity relation is the relation of equivalence so it is reflexive, symmetric and transitive relation.Unfortunately I don't know what to use this information in my task.
Can you get some tips?
linear-algebra
asked Feb 2 at 22:35
VirtualUserVirtualUser
1,321317
$endgroup$
add a comment |
$begingroup$
Triangular matrix $M in mathbb R^{n,n}$ and all elements on the diagonal are different. Proof that $M$ and $M^{T}$ are similar.
I know that the matrices are similar when the matrix similarity relation is the relation of equivalence so it is reflexive, symmetric and transitive relation.Unfortunately I don't know what to use this information in my task.
Can you get some tips?
linear-algebra
asked Feb 2 at 22:35
VirtualUserVirtualUser
1,321317
$endgroup$
3
$begingroup$
$M$ is diagonalizable, that is $P^{-1}MP = D$
$endgroup$
– Will Jagy
Feb 2 at 22:40
add a comment |
$begingroup$
Triangular matrix $M in mathbb R^{n,n}$ and all elements on the diagonal are different. Proof that $M$ and $M^{T}$ are similar.
I know that the matrices are similar when the matrix similarity relation is the relation of equivalence so it is reflexive, symmetric and transitive relation.Unfortunately I don't know what to use this information in my task.
Can you get some tips?
linear-algebra
asked Feb 2 at 22:35
VirtualUserVirtualUser
1,321317
$endgroup$
Triangular matrix $M in mathbb R^{n,n}$ and all elements on the diagonal are different. Proof that $M$ and $M^{T}$ are similar.
I know that the matrices are similar when the matrix similarity relation is the relation of equivalence so it is reflexive, symmetric and transitive relation.Unfortunately I don't know what to use this information in my task.
Can you get some tips?
linear-algebra
linear-algebra
asked Feb 2 at 22:35
VirtualUserVirtualUser
1,321317
asked Feb 2 at 22:35
VirtualUserVirtualUser
1,321317
asked Feb 2 at 22:35
VirtualUserVirtualUser
1,321317
asked Feb 2 at 22:35
VirtualUserVirtualUser
1,321317
asked Feb 2 at 22:35
VirtualUserVirtualUser
1,321317
1,321317
3
$begingroup$
$M$ is diagonalizable, that is $P^{-1}MP = D$
$endgroup$
– Will Jagy
Feb 2 at 22:40
add a comment |
3
$begingroup$
$M$ is diagonalizable, that is $P^{-1}MP = D$
$endgroup$
– Will Jagy
Feb 2 at 22:40
3
3
$begingroup$
$M$ is diagonalizable, that is $P^{-1}MP = D$
$endgroup$
– Will Jagy
Feb 2 at 22:40
$begingroup$
$M$ is diagonalizable, that is $P^{-1}MP = D$
$endgroup$
– Will Jagy
Feb 2 at 22:40
add a comment |
1 Answer
1
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$begingroup$
Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?
Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).
answered Feb 2 at 22:44
Jacob MaibachJacob Maibach
1,4902917
$endgroup$
add a comment |
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$begingroup$
Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?
Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).
answered Feb 2 at 22:44
Jacob MaibachJacob Maibach
1,4902917
$endgroup$
add a comment |
$begingroup$
Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?
Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).
answered Feb 2 at 22:44
Jacob MaibachJacob Maibach
1,4902917
$endgroup$
add a comment |
$begingroup$
Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?
Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).
answered Feb 2 at 22:44
Jacob MaibachJacob Maibach
1,4902917
$endgroup$
Presume that they are similar, and in particular $M = P^{-1}M^{T}P$. What can you say about $P$?
Then write the equation as $PM = M^{T}P$. Given what we know about $P$, this should suggest what we need for $M$ and $M^{T}$ to be similar (i.e. it should tell us what type of matrix $P$ is sufficient to make the equation true).
answered Feb 2 at 22:44
Jacob MaibachJacob Maibach
1,4902917
answered Feb 2 at 22:44
Jacob MaibachJacob Maibach
1,4902917
answered Feb 2 at 22:44
Jacob MaibachJacob Maibach
1,4902917
answered Feb 2 at 22:44
Jacob MaibachJacob Maibach
1,4902917
1,4902917
add a comment |
add a comment |
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$begingroup$
$M$ is diagonalizable, that is $P^{-1}MP = D$
$endgroup$
– Will Jagy
Feb 2 at 22:40