What transparent parametrizations of 2x2 matrices are there? The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 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
Is there a writing software that you can sort scenes like slides in PowerPoint?
Why does the Event Horizon Telescope (EHT) not include telescopes from Africa, Asia or Australia?
How do I add random spotting to the same face in cycles?
Relations between two reciprocal partial derivatives?
Why can't devices on different VLANs, but on the same subnet, communicate?
What was the last x86 CPU that did not have the x87 floating-point unit built in?
What can I do if neighbor is blocking my solar panels intentionally?
Segmentation fault output is suppressed when piping stdin into a function. Why?
What are these Gizmos at Izaña Atmospheric Research Center in Spain?
Is every episode of "Where are my Pants?" identical?
In horse breeding, what is the female equivalent of putting a horse out "to stud"?
What is special about square numbers here?
Make it rain characters
University's motivation for having tenure-track positions
Keeping a retro style to sci-fi spaceships?
What information about me do stores get via my credit card?
First use of “packing” as in carrying a gun
Is it ethical to upload a automatically generated paper to a non peer-reviewed site as part of a larger research?
Do warforged have souls?
Can a 1st-level character have an ability score above 18?
ELI5: Why do they say that Israel would have been the fourth country to land a spacecraft on the Moon and why do they call it low cost?
How did the audience guess the pentatonic scale in Bobby McFerrin's presentation?
Can withdrawing asylum be illegal?
Why is superheterodyning better than direct conversion?
What transparent parametrizations of 2x2 matrices are there?
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 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.
$$ A_a,b,c,d = beginbmatrix
a& b\
c & d
endbmatrix$$
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.
$$ B_lambda_1,lambda_2 = beginbmatrix
lambda_1& 0\
0 & lambda_2
endbmatrix$$
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.
$$ C_zeta,omega = beginbmatrix
zeta& -omega\
omega & zeta
endbmatrix
$$
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
$endgroup$
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.
$$ A_a,b,c,d = beginbmatrix
a& b\
c & d
endbmatrix$$
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.
$$ B_lambda_1,lambda_2 = beginbmatrix
lambda_1& 0\
0 & lambda_2
endbmatrix$$
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.
$$ C_zeta,omega = beginbmatrix
zeta& -omega\
omega & zeta
endbmatrix
$$
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
$endgroup$
$begingroup$
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: $$pmatrixa&0cr0&bcr,pmatrixa&1cr0&acr,pmatrixa&-bcr b&a$$ 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.
$$ A_a,b,c,d = beginbmatrix
a& b\
c & d
endbmatrix$$
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.
$$ B_lambda_1,lambda_2 = beginbmatrix
lambda_1& 0\
0 & lambda_2
endbmatrix$$
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.
$$ C_zeta,omega = beginbmatrix
zeta& -omega\
omega & zeta
endbmatrix
$$
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
$endgroup$
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.
$$ A_a,b,c,d = beginbmatrix
a& b\
c & d
endbmatrix$$
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.
$$ B_lambda_1,lambda_2 = beginbmatrix
lambda_1& 0\
0 & lambda_2
endbmatrix$$
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.
$$ C_zeta,omega = beginbmatrix
zeta& -omega\
omega & zeta
endbmatrix
$$
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
edited Mar 25 at 10:19
LudvigH
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: $$pmatrixa&0cr0&bcr,pmatrixa&1cr0&acr,pmatrixa&-bcr b&a$$ 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: $$pmatrixa&0cr0&bcr,pmatrixa&1cr0&acr,pmatrixa&-bcr b&a$$ 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: $$pmatrixa&0cr0&bcr,pmatrixa&1cr0&acr,pmatrixa&-bcr b&a$$ 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: $$pmatrixa&0cr0&bcr,pmatrixa&1cr0&acr,pmatrixa&-bcr b&a$$ 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: $$pmatrixa&0cr0&bcr,quadpmatrixa&1cr0&acr,quadpmatrixa&-bcr b&acr$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.
$endgroup$
$begingroup$
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 |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
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%2f3161538%2fwhat-transparent-parametrizations-of-2x2-matrices-are-there%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrixa&0cr0&bcr,quadpmatrixa&1cr0&acr,quadpmatrixa&-bcr b&acr$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.
$endgroup$
$begingroup$
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: $$pmatrixa&0cr0&bcr,quadpmatrixa&1cr0&acr,quadpmatrixa&-bcr b&acr$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.
$endgroup$
$begingroup$
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: $$pmatrixa&0cr0&bcr,quadpmatrixa&1cr0&acr,quadpmatrixa&-bcr b&acr$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.
$endgroup$
Every $2times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$pmatrixa&0cr0&bcr,quadpmatrixa&1cr0&acr,quadpmatrixa&-bcr b&acr$$ 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
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 |
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.
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%2f3161538%2fwhat-transparent-parametrizations-of-2x2-matrices-are-there%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
$begingroup$
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: $$pmatrixa&0cr0&bcr,pmatrixa&1cr0&acr,pmatrixa&-bcr b&a$$ 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