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What transparent parametrizations of 2x2 matrices are there?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Solution to system of difference equations with repeated unit rootsEigenvectors of a $2 times 2$ matrix when the eigenvalues are not integersB and C matrices for real modal representation of a 2x2 linear system with complex eigenvaluesInverse spectrum problem - showing the existence of a 2x2 doubly stochastic matrix,Prove that $lambda_1^2$, $lambda_1lambda_2$ and $lambda_2^2$ are eigenvalues of matrix $A$Find “REAL” cannonical form of 4x4 matrixConjugate classes for 2x2 matriceseigenvalues of a 6x6 matrix with many zerosDetermining the maximum number of linearly independent rows and columns for a given matrixEquilibria of 2x2 linear system
$begingroup$
I want to contruct a time series by a recurrence relation $y_t = Ay_{t-1}$, where A is a 2x2 real matrix. Assume $y_0$ is some known initial condition. Also, I assume $y_i in mathbb R^2$.
To understand the system without having to solve the equation, it would be nice to have a parametrization of the matrix that is transparent with the behavior of the solution.
The most simple parametrization would be by the real entries in the matrix. Use $(a,b,c,d)$. Unfortunately, it is not easy to understand the dynamics of the system easily by this set of parameters.
$$ A_{a,b,c,d} = begin{bmatrix}
a& b\
c & d
end{bmatrix}$$
Option two is if the matrix is (real) diagonalizable. Taking the eigenvalues $(lambda_1,lambda_2)$. On the downside, this will only give system dynamics that are exponential growth of decline.
$$ B_{lambda_1,lambda_2} = begin{bmatrix}
lambda_1& 0\
0 & lambda_2
end{bmatrix}$$
Option three is to think of the solutions as damped pendulum solutions. Then the paramterization is $(zeta, omega)$, giving below matrix. The notation is inspired by https://en.wikipedia.org/wiki/Harmonic_oscillator, but it is not perfectly equivalent. In this notation, I can easily read off the dampening of the solutions, since $zeta$ is the real part of the eigenvalues, and the frequency, since $omega$ gives the imaginary part of the eigenvalues.
$$ C_{zeta,omega} = begin{bmatrix}
zeta& -omega\
omega & zeta
end{bmatrix}
$$
The three examples are not satisfactory to me, since they produce either (A) solutions that are not easily coupled to the parameters, (B) noninteresting dynamics, or (C) only spans a 2-dimensional subspace of all 2x2-matrices.
My question: What other parametrizations of real 2x2 matrices is there, that give instant intuition for the corresponding dynamical system?
matrices dynamical-systems
asked Mar 25 at 8:59
LudvigHLudvigH
1186
$endgroup$
add a comment |
$begingroup$
I want to contruct a time series by a recurrence relation $y_t = Ay_{t-1}$, where A is a 2x2 real matrix. Assume $y_0$ is some known initial condition. Also, I assume $y_i in mathbb R^2$.
To understand the system without having to solve the equation, it would be nice to have a parametrization of the matrix that is transparent with the behavior of the solution.
The most simple parametrization would be by the real entries in the matrix. Use $(a,b,c,d)$. Unfortunately, it is not easy to understand the dynamics of the system easily by this set of parameters.
$$ A_{a,b,c,d} = begin{bmatrix}
a& b\
c & d
end{bmatrix}$$
Option two is if the matrix is (real) diagonalizable. Taking the eigenvalues $(lambda_1,lambda_2)$. On the downside, this will only give system dynamics that are exponential growth of decline.
$$ B_{lambda_1,lambda_2} = begin{bmatrix}
lambda_1& 0\
0 & lambda_2
end{bmatrix}$$
Option three is to think of the solutions as damped pendulum solutions. Then the paramterization is $(zeta, omega)$, giving below matrix. The notation is inspired by https://en.wikipedia.org/wiki/Harmonic_oscillator, but it is not perfectly equivalent. In this notation, I can easily read off the dampening of the solutions, since $zeta$ is the real part of the eigenvalues, and the frequency, since $omega$ gives the imaginary part of the eigenvalues.
$$ C_{zeta,omega} = begin{bmatrix}
zeta& -omega\
omega & zeta
end{bmatrix}
$$
The three examples are not satisfactory to me, since they produce either (A) solutions that are not easily coupled to the parameters, (B) noninteresting dynamics, or (C) only spans a 2-dimensional subspace of all 2x2-matrices.
My question: What other parametrizations of real 2x2 matrices is there, that give instant intuition for the corresponding dynamical system?
matrices dynamical-systems
asked Mar 25 at 8:59
LudvigHLudvigH
1186
$endgroup$
$begingroup$
I don't know what you mean by parametrization of a matrix, but, if the matrix is diagonalizable, then the recurrence can be rewritten in terms of the diagonal matrix (and the eigenvectors).
$endgroup$
– Gerry Myerson
Mar 25 at 9:53
$begingroup$
I have updated the question now. Is it clearer in this phrasing?
$endgroup$
– LudvigH
Mar 25 at 10:20
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},pmatrix{a&1cr0&acr},pmatrix{a&-bcr b&a}$$ Each of those gives about as much insight as there is into the behavior of the dynamical system.
$endgroup$
– Gerry Myerson
Mar 25 at 10:40
$begingroup$
That is just about the type of answer I hoped for. Write it as an answer and I'll accept it. :)
$endgroup$
– LudvigH
Mar 25 at 11:14
add a comment |
$begingroup$
I want to contruct a time series by a recurrence relation $y_t = Ay_{t-1}$, where A is a 2x2 real matrix. Assume $y_0$ is some known initial condition. Also, I assume $y_i in mathbb R^2$.
To understand the system without having to solve the equation, it would be nice to have a parametrization of the matrix that is transparent with the behavior of the solution.
The most simple parametrization would be by the real entries in the matrix. Use $(a,b,c,d)$. Unfortunately, it is not easy to understand the dynamics of the system easily by this set of parameters.
$$ A_{a,b,c,d} = begin{bmatrix}
a& b\
c & d
end{bmatrix}$$
Option two is if the matrix is (real) diagonalizable. Taking the eigenvalues $(lambda_1,lambda_2)$. On the downside, this will only give system dynamics that are exponential growth of decline.
$$ B_{lambda_1,lambda_2} = begin{bmatrix}
lambda_1& 0\
0 & lambda_2
end{bmatrix}$$
Option three is to think of the solutions as damped pendulum solutions. Then the paramterization is $(zeta, omega)$, giving below matrix. The notation is inspired by https://en.wikipedia.org/wiki/Harmonic_oscillator, but it is not perfectly equivalent. In this notation, I can easily read off the dampening of the solutions, since $zeta$ is the real part of the eigenvalues, and the frequency, since $omega$ gives the imaginary part of the eigenvalues.
$$ C_{zeta,omega} = begin{bmatrix}
zeta& -omega\
omega & zeta
end{bmatrix}
$$
The three examples are not satisfactory to me, since they produce either (A) solutions that are not easily coupled to the parameters, (B) noninteresting dynamics, or (C) only spans a 2-dimensional subspace of all 2x2-matrices.
My question: What other parametrizations of real 2x2 matrices is there, that give instant intuition for the corresponding dynamical system?
matrices dynamical-systems
asked Mar 25 at 8:59
LudvigHLudvigH
1186
$endgroup$
I want to contruct a time series by a recurrence relation $y_t = Ay_{t-1}$, where A is a 2x2 real matrix. Assume $y_0$ is some known initial condition. Also, I assume $y_i in mathbb R^2$.
To understand the system without having to solve the equation, it would be nice to have a parametrization of the matrix that is transparent with the behavior of the solution.
The most simple parametrization would be by the real entries in the matrix. Use $(a,b,c,d)$. Unfortunately, it is not easy to understand the dynamics of the system easily by this set of parameters.
$$ A_{a,b,c,d} = begin{bmatrix}
a& b\
c & d
end{bmatrix}$$
Option two is if the matrix is (real) diagonalizable. Taking the eigenvalues $(lambda_1,lambda_2)$. On the downside, this will only give system dynamics that are exponential growth of decline.
$$ B_{lambda_1,lambda_2} = begin{bmatrix}
lambda_1& 0\
0 & lambda_2
end{bmatrix}$$
Option three is to think of the solutions as damped pendulum solutions. Then the paramterization is $(zeta, omega)$, giving below matrix. The notation is inspired by https://en.wikipedia.org/wiki/Harmonic_oscillator, but it is not perfectly equivalent. In this notation, I can easily read off the dampening of the solutions, since $zeta$ is the real part of the eigenvalues, and the frequency, since $omega$ gives the imaginary part of the eigenvalues.
$$ C_{zeta,omega} = begin{bmatrix}
zeta& -omega\
omega & zeta
end{bmatrix}
$$
The three examples are not satisfactory to me, since they produce either (A) solutions that are not easily coupled to the parameters, (B) noninteresting dynamics, or (C) only spans a 2-dimensional subspace of all 2x2-matrices.
My question: What other parametrizations of real 2x2 matrices is there, that give instant intuition for the corresponding dynamical system?
matrices dynamical-systems
matrices dynamical-systems
asked Mar 25 at 8:59
LudvigHLudvigH
1186
asked Mar 25 at 8:59
LudvigHLudvigH
1186
edited Mar 25 at 10:19
LudvigH
asked Mar 25 at 8:59
LudvigHLudvigH
1186
asked Mar 25 at 8:59
LudvigHLudvigH
1186
asked Mar 25 at 8:59
LudvigHLudvigH
1186
1186
$begingroup$
I don't know what you mean by parametrization of a matrix, but, if the matrix is diagonalizable, then the recurrence can be rewritten in terms of the diagonal matrix (and the eigenvectors).
$endgroup$
– Gerry Myerson
Mar 25 at 9:53
$begingroup$
I have updated the question now. Is it clearer in this phrasing?
$endgroup$
– LudvigH
Mar 25 at 10:20
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},pmatrix{a&1cr0&acr},pmatrix{a&-bcr b&a}$$ Each of those gives about as much insight as there is into the behavior of the dynamical system.
$endgroup$
– Gerry Myerson
Mar 25 at 10:40
$begingroup$
That is just about the type of answer I hoped for. Write it as an answer and I'll accept it. :)
$endgroup$
– LudvigH
Mar 25 at 11:14
add a comment |
$begingroup$
I don't know what you mean by parametrization of a matrix, but, if the matrix is diagonalizable, then the recurrence can be rewritten in terms of the diagonal matrix (and the eigenvectors).
$endgroup$
– Gerry Myerson
Mar 25 at 9:53
$begingroup$
I have updated the question now. Is it clearer in this phrasing?
$endgroup$
– LudvigH
Mar 25 at 10:20
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},pmatrix{a&1cr0&acr},pmatrix{a&-bcr b&a}$$ Each of those gives about as much insight as there is into the behavior of the dynamical system.
$endgroup$
– Gerry Myerson
Mar 25 at 10:40
$begingroup$
That is just about the type of answer I hoped for. Write it as an answer and I'll accept it. :)
$endgroup$
– LudvigH
Mar 25 at 11:14
$begingroup$
I don't know what you mean by parametrization of a matrix, but, if the matrix is diagonalizable, then the recurrence can be rewritten in terms of the diagonal matrix (and the eigenvectors).
$endgroup$
– Gerry Myerson
Mar 25 at 9:53
$begingroup$
I don't know what you mean by parametrization of a matrix, but, if the matrix is diagonalizable, then the recurrence can be rewritten in terms of the diagonal matrix (and the eigenvectors).
$endgroup$
– Gerry Myerson
Mar 25 at 9:53
$begingroup$
I have updated the question now. Is it clearer in this phrasing?
$endgroup$
– LudvigH
Mar 25 at 10:20
$begingroup$
I have updated the question now. Is it clearer in this phrasing?
$endgroup$
– LudvigH
Mar 25 at 10:20
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},pmatrix{a&1cr0&acr},pmatrix{a&-bcr b&a}$$ Each of those gives about as much insight as there is into the behavior of the dynamical system.
$endgroup$
– Gerry Myerson
Mar 25 at 10:40
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},pmatrix{a&1cr0&acr},pmatrix{a&-bcr b&a}$$ Each of those gives about as much insight as there is into the behavior of the dynamical system.
$endgroup$
– Gerry Myerson
Mar 25 at 10:40
$begingroup$
That is just about the type of answer I hoped for. Write it as an answer and I'll accept it. :)
$endgroup$
– LudvigH
Mar 25 at 11:14
$begingroup$
That is just about the type of answer I hoped for. Write it as an answer and I'll accept it. :)
$endgroup$
– LudvigH
Mar 25 at 11:14
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},quadpmatrix{a&1cr0&acr},quadpmatrix{a&-bcr b&acr}$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.
answered Mar 25 at 11:34
Gerry MyersonGerry Myerson
148k8152306
$endgroup$
$begingroup$
So what do you call the three? Diagonal form, Jordan form and 'real jordan form' ?
$endgroup$
– LudvigH
Mar 25 at 12:43
1
$begingroup$
The third one could be called real canonical form as at math.byu.edu/~grant/courses/m634/f99/lec11.pdf Diagonal form is just a special case of Jordan form.
$endgroup$
– Gerry Myerson
Mar 25 at 21:58
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},quadpmatrix{a&1cr0&acr},quadpmatrix{a&-bcr b&acr}$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.
answered Mar 25 at 11:34
Gerry MyersonGerry Myerson
148k8152306
$endgroup$
$begingroup$
So what do you call the three? Diagonal form, Jordan form and 'real jordan form' ?
$endgroup$
– LudvigH
Mar 25 at 12:43
1
$begingroup$
The third one could be called real canonical form as at math.byu.edu/~grant/courses/m634/f99/lec11.pdf Diagonal form is just a special case of Jordan form.
$endgroup$
– Gerry Myerson
Mar 25 at 21:58
add a comment |
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},quadpmatrix{a&1cr0&acr},quadpmatrix{a&-bcr b&acr}$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.
answered Mar 25 at 11:34
Gerry MyersonGerry Myerson
148k8152306
$endgroup$
$begingroup$
So what do you call the three? Diagonal form, Jordan form and 'real jordan form' ?
$endgroup$
– LudvigH
Mar 25 at 12:43
1
$begingroup$
The third one could be called real canonical form as at math.byu.edu/~grant/courses/m634/f99/lec11.pdf Diagonal form is just a special case of Jordan form.
$endgroup$
– Gerry Myerson
Mar 25 at 21:58
add a comment |
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},quadpmatrix{a&1cr0&acr},quadpmatrix{a&-bcr b&acr}$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.
answered Mar 25 at 11:34
Gerry MyersonGerry Myerson
148k8152306
$endgroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},quadpmatrix{a&1cr0&acr},quadpmatrix{a&-bcr b&acr}$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.
answered Mar 25 at 11:34
Gerry MyersonGerry Myerson
148k8152306
answered Mar 25 at 11:34
Gerry MyersonGerry Myerson
148k8152306
answered Mar 25 at 11:34
Gerry MyersonGerry Myerson
148k8152306
answered Mar 25 at 11:34
Gerry MyersonGerry Myerson
148k8152306
148k8152306
$begingroup$
So what do you call the three? Diagonal form, Jordan form and 'real jordan form' ?
$endgroup$
– LudvigH
Mar 25 at 12:43
1
$begingroup$
The third one could be called real canonical form as at math.byu.edu/~grant/courses/m634/f99/lec11.pdf Diagonal form is just a special case of Jordan form.
$endgroup$
– Gerry Myerson
Mar 25 at 21:58
add a comment |
$begingroup$
So what do you call the three? Diagonal form, Jordan form and 'real jordan form' ?
$endgroup$
– LudvigH
Mar 25 at 12:43
1
$begingroup$
The third one could be called real canonical form as at math.byu.edu/~grant/courses/m634/f99/lec11.pdf Diagonal form is just a special case of Jordan form.
$endgroup$
– Gerry Myerson
Mar 25 at 21:58
$begingroup$
So what do you call the three? Diagonal form, Jordan form and 'real jordan form' ?
$endgroup$
– LudvigH
Mar 25 at 12:43
$begingroup$
So what do you call the three? Diagonal form, Jordan form and 'real jordan form' ?
$endgroup$
– LudvigH
Mar 25 at 12:43
1
1
$begingroup$
The third one could be called real canonical form as at math.byu.edu/~grant/courses/m634/f99/lec11.pdf Diagonal form is just a special case of Jordan form.
$endgroup$
– Gerry Myerson
Mar 25 at 21:58
$begingroup$
The third one could be called real canonical form as at math.byu.edu/~grant/courses/m634/f99/lec11.pdf Diagonal form is just a special case of Jordan form.
$endgroup$
– Gerry Myerson
Mar 25 at 21:58
add a comment |
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$begingroup$
I don't know what you mean by parametrization of a matrix, but, if the matrix is diagonalizable, then the recurrence can be rewritten in terms of the diagonal matrix (and the eigenvectors).
$endgroup$
– Gerry Myerson
Mar 25 at 9:53
$begingroup$
I have updated the question now. Is it clearer in this phrasing?
$endgroup$
– LudvigH
Mar 25 at 10:20
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrix{a&0cr0&bcr},pmatrix{a&1cr0&acr},pmatrix{a&-bcr b&a}$$ Each of those gives about as much insight as there is into the behavior of the dynamical system.
$endgroup$
– Gerry Myerson
Mar 25 at 10:40
$begingroup$
That is just about the type of answer I hoped for. Write it as an answer and I'll accept it. :)
$endgroup$
– LudvigH
Mar 25 at 11:14