Solution of a linear matrix differential equation
up vote
6
down vote
favorite
Consider a linear matrix differential equation of the form
$$frac{mathrm{d} C}{mathrm{d} t} = A C + C A^{mathrm{T}}$$
where $C$ is a symmetric $n times n$ matrix and $A$ is a $n times n$ matrix. Find $C(t)$.
Is there a formal solution for the above equation? This is in principle linear equation if we treat the matrix $C$ and $A$ as a $n^2$ vector. However, it does not seem to be practical way to solve the problem.
This kind of differential equations for matrices is quite new to me.
Besides the formal solution let me know some books considering similar topic. Thanks.
differential-equations
edited Nov 17 at 21:40
Rodrigo de Azevedo
12.8k41753
asked Nov 23 '13 at 15:41
Sungmin
240210
add a comment |
up vote
6
down vote
favorite
Consider a linear matrix differential equation of the form
$$frac{mathrm{d} C}{mathrm{d} t} = A C + C A^{mathrm{T}}$$
where $C$ is a symmetric $n times n$ matrix and $A$ is a $n times n$ matrix. Find $C(t)$.
Is there a formal solution for the above equation? This is in principle linear equation if we treat the matrix $C$ and $A$ as a $n^2$ vector. However, it does not seem to be practical way to solve the problem.
This kind of differential equations for matrices is quite new to me.
Besides the formal solution let me know some books considering similar topic. Thanks.
differential-equations
edited Nov 17 at 21:40
Rodrigo de Azevedo
12.8k41753
asked Nov 23 '13 at 15:41
Sungmin
240210
add a comment |
up vote
6
down vote
favorite
up vote
6
down vote
favorite
Consider a linear matrix differential equation of the form
$$frac{mathrm{d} C}{mathrm{d} t} = A C + C A^{mathrm{T}}$$
where $C$ is a symmetric $n times n$ matrix and $A$ is a $n times n$ matrix. Find $C(t)$.
Is there a formal solution for the above equation? This is in principle linear equation if we treat the matrix $C$ and $A$ as a $n^2$ vector. However, it does not seem to be practical way to solve the problem.
This kind of differential equations for matrices is quite new to me.
Besides the formal solution let me know some books considering similar topic. Thanks.
differential-equations
edited Nov 17 at 21:40
Rodrigo de Azevedo
12.8k41753
asked Nov 23 '13 at 15:41
Sungmin
240210
Consider a linear matrix differential equation of the form
$$frac{mathrm{d} C}{mathrm{d} t} = A C + C A^{mathrm{T}}$$
where $C$ is a symmetric $n times n$ matrix and $A$ is a $n times n$ matrix. Find $C(t)$.
Is there a formal solution for the above equation? This is in principle linear equation if we treat the matrix $C$ and $A$ as a $n^2$ vector. However, it does not seem to be practical way to solve the problem.
This kind of differential equations for matrices is quite new to me.
Besides the formal solution let me know some books considering similar topic. Thanks.
differential-equations
differential-equations
edited Nov 17 at 21:40
Rodrigo de Azevedo
12.8k41753
asked Nov 23 '13 at 15:41
Sungmin
240210
edited Nov 17 at 21:40
Rodrigo de Azevedo
12.8k41753
asked Nov 23 '13 at 15:41
Sungmin
240210
edited Nov 17 at 21:40
Rodrigo de Azevedo
12.8k41753
edited Nov 17 at 21:40
Rodrigo de Azevedo
12.8k41753
edited Nov 17 at 21:40
Rodrigo de Azevedo
12.8k41753
12.8k41753
asked Nov 23 '13 at 15:41
Sungmin
240210
asked Nov 23 '13 at 15:41
Sungmin
240210
asked Nov 23 '13 at 15:41
Sungmin
240210
240210
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
You can solve the differential equation as follows: write
$$frac{dC}{dt} = AC+CA^T = left(Aotimes1+1otimes A^Tright)C$$
This gives us the solution
$$C(t)=expleft(tleft(Aotimes1+1otimes A^Tright)right)C(0)$$
Notice that this is basically solving by looking at $A$ as a $n^2$-vector, as you said in the question. The nice thing with this formulation is that you notice at once that $Aotimes1$ and $1otimes A^T$ commute, and thus we have
$$expleft(tleft(Aotimes1+1otimes A^Tright)right) = expleft(tleft(Aotimes1right)right)expleft(tleft(1otimes A^Tright)right)=$$
$$=left(exp(tA)otimes1right)left(1otimes expleft(tA^Tright)right)$$
So
$$C(t) = exp(tA)C(0)expleft(tAright)^T$$
which you can easily check to be correct.
answered Nov 23 '13 at 16:14
Daniel Robert-Nicoud
20.2k33596
In the first equation, what is the dimension of identity matrix?
– Sungmin
Nov 24 '13 at 0:15
@Sungmin It must be $ntimes n$ for everything to make sense.
– Daniel Robert-Nicoud
Nov 24 '13 at 1:05
There seems to be an error somewhere here, the dimensions don't match in several matrices being claim as equal. Assuming by $otimes$ you mean the kronecker product, $(A otimes I)C$ has a larger dimension than AC for example. As square matrices, $Dim(A otimes I)=Dim(A)Dim(I) = n^2$ but $Dim(AC)=n$.
– Benjamin
Aug 12 '15 at 20:42
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
You can solve the differential equation as follows: write
$$frac{dC}{dt} = AC+CA^T = left(Aotimes1+1otimes A^Tright)C$$
This gives us the solution
$$C(t)=expleft(tleft(Aotimes1+1otimes A^Tright)right)C(0)$$
Notice that this is basically solving by looking at $A$ as a $n^2$-vector, as you said in the question. The nice thing with this formulation is that you notice at once that $Aotimes1$ and $1otimes A^T$ commute, and thus we have
$$expleft(tleft(Aotimes1+1otimes A^Tright)right) = expleft(tleft(Aotimes1right)right)expleft(tleft(1otimes A^Tright)right)=$$
$$=left(exp(tA)otimes1right)left(1otimes expleft(tA^Tright)right)$$
So
$$C(t) = exp(tA)C(0)expleft(tAright)^T$$
which you can easily check to be correct.
answered Nov 23 '13 at 16:14
Daniel Robert-Nicoud
20.2k33596
In the first equation, what is the dimension of identity matrix?
– Sungmin
Nov 24 '13 at 0:15
@Sungmin It must be $ntimes n$ for everything to make sense.
– Daniel Robert-Nicoud
Nov 24 '13 at 1:05
There seems to be an error somewhere here, the dimensions don't match in several matrices being claim as equal. Assuming by $otimes$ you mean the kronecker product, $(A otimes I)C$ has a larger dimension than AC for example. As square matrices, $Dim(A otimes I)=Dim(A)Dim(I) = n^2$ but $Dim(AC)=n$.
– Benjamin
Aug 12 '15 at 20:42
add a comment |
up vote
1
down vote
accepted
You can solve the differential equation as follows: write
$$frac{dC}{dt} = AC+CA^T = left(Aotimes1+1otimes A^Tright)C$$
This gives us the solution
$$C(t)=expleft(tleft(Aotimes1+1otimes A^Tright)right)C(0)$$
Notice that this is basically solving by looking at $A$ as a $n^2$-vector, as you said in the question. The nice thing with this formulation is that you notice at once that $Aotimes1$ and $1otimes A^T$ commute, and thus we have
$$expleft(tleft(Aotimes1+1otimes A^Tright)right) = expleft(tleft(Aotimes1right)right)expleft(tleft(1otimes A^Tright)right)=$$
$$=left(exp(tA)otimes1right)left(1otimes expleft(tA^Tright)right)$$
So
$$C(t) = exp(tA)C(0)expleft(tAright)^T$$
which you can easily check to be correct.
answered Nov 23 '13 at 16:14
Daniel Robert-Nicoud
20.2k33596
In the first equation, what is the dimension of identity matrix?
– Sungmin
Nov 24 '13 at 0:15
@Sungmin It must be $ntimes n$ for everything to make sense.
– Daniel Robert-Nicoud
Nov 24 '13 at 1:05
There seems to be an error somewhere here, the dimensions don't match in several matrices being claim as equal. Assuming by $otimes$ you mean the kronecker product, $(A otimes I)C$ has a larger dimension than AC for example. As square matrices, $Dim(A otimes I)=Dim(A)Dim(I) = n^2$ but $Dim(AC)=n$.
– Benjamin
Aug 12 '15 at 20:42
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
You can solve the differential equation as follows: write
$$frac{dC}{dt} = AC+CA^T = left(Aotimes1+1otimes A^Tright)C$$
This gives us the solution
$$C(t)=expleft(tleft(Aotimes1+1otimes A^Tright)right)C(0)$$
Notice that this is basically solving by looking at $A$ as a $n^2$-vector, as you said in the question. The nice thing with this formulation is that you notice at once that $Aotimes1$ and $1otimes A^T$ commute, and thus we have
$$expleft(tleft(Aotimes1+1otimes A^Tright)right) = expleft(tleft(Aotimes1right)right)expleft(tleft(1otimes A^Tright)right)=$$
$$=left(exp(tA)otimes1right)left(1otimes expleft(tA^Tright)right)$$
So
$$C(t) = exp(tA)C(0)expleft(tAright)^T$$
which you can easily check to be correct.
answered Nov 23 '13 at 16:14
Daniel Robert-Nicoud
20.2k33596
You can solve the differential equation as follows: write
$$frac{dC}{dt} = AC+CA^T = left(Aotimes1+1otimes A^Tright)C$$
This gives us the solution
$$C(t)=expleft(tleft(Aotimes1+1otimes A^Tright)right)C(0)$$
Notice that this is basically solving by looking at $A$ as a $n^2$-vector, as you said in the question. The nice thing with this formulation is that you notice at once that $Aotimes1$ and $1otimes A^T$ commute, and thus we have
$$expleft(tleft(Aotimes1+1otimes A^Tright)right) = expleft(tleft(Aotimes1right)right)expleft(tleft(1otimes A^Tright)right)=$$
$$=left(exp(tA)otimes1right)left(1otimes expleft(tA^Tright)right)$$
So
$$C(t) = exp(tA)C(0)expleft(tAright)^T$$
which you can easily check to be correct.
answered Nov 23 '13 at 16:14
Daniel Robert-Nicoud
20.2k33596
edited Nov 23 '13 at 17:52
answered Nov 23 '13 at 16:14
Daniel Robert-Nicoud
20.2k33596
answered Nov 23 '13 at 16:14
Daniel Robert-Nicoud
20.2k33596
answered Nov 23 '13 at 16:14
Daniel Robert-Nicoud
20.2k33596
20.2k33596
In the first equation, what is the dimension of identity matrix?
– Sungmin
Nov 24 '13 at 0:15
@Sungmin It must be $ntimes n$ for everything to make sense.
– Daniel Robert-Nicoud
Nov 24 '13 at 1:05
There seems to be an error somewhere here, the dimensions don't match in several matrices being claim as equal. Assuming by $otimes$ you mean the kronecker product, $(A otimes I)C$ has a larger dimension than AC for example. As square matrices, $Dim(A otimes I)=Dim(A)Dim(I) = n^2$ but $Dim(AC)=n$.
– Benjamin
Aug 12 '15 at 20:42
add a comment |
In the first equation, what is the dimension of identity matrix?
– Sungmin
Nov 24 '13 at 0:15
@Sungmin It must be $ntimes n$ for everything to make sense.
– Daniel Robert-Nicoud
Nov 24 '13 at 1:05
There seems to be an error somewhere here, the dimensions don't match in several matrices being claim as equal. Assuming by $otimes$ you mean the kronecker product, $(A otimes I)C$ has a larger dimension than AC for example. As square matrices, $Dim(A otimes I)=Dim(A)Dim(I) = n^2$ but $Dim(AC)=n$.
– Benjamin
Aug 12 '15 at 20:42
In the first equation, what is the dimension of identity matrix?
– Sungmin
Nov 24 '13 at 0:15
In the first equation, what is the dimension of identity matrix?
– Sungmin
Nov 24 '13 at 0:15
@Sungmin It must be $ntimes n$ for everything to make sense.
– Daniel Robert-Nicoud
Nov 24 '13 at 1:05
@Sungmin It must be $ntimes n$ for everything to make sense.
– Daniel Robert-Nicoud
Nov 24 '13 at 1:05
There seems to be an error somewhere here, the dimensions don't match in several matrices being claim as equal. Assuming by $otimes$ you mean the kronecker product, $(A otimes I)C$ has a larger dimension than AC for example. As square matrices, $Dim(A otimes I)=Dim(A)Dim(I) = n^2$ but $Dim(AC)=n$.
– Benjamin
Aug 12 '15 at 20:42
There seems to be an error somewhere here, the dimensions don't match in several matrices being claim as equal. Assuming by $otimes$ you mean the kronecker product, $(A otimes I)C$ has a larger dimension than AC for example. As square matrices, $Dim(A otimes I)=Dim(A)Dim(I) = n^2$ but $Dim(AC)=n$.
– Benjamin
Aug 12 '15 at 20:42
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f578238%2fsolution-of-a-linear-matrix-differential-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
