Transforming back and forth between reference frames using orthogonal transformation matrices Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Q: Bases and Frames using Fourier SeriesQuestion about orthogonal transformation / orthogonal matricesHow can I interpret $2 times 2$ and $3 times 3$ transformation matrices geometrically?How to compute determinant (or eigenvalues) of this matrix?Properties of orthogonal, singular and antisymmetric matrices.Linear transformation matrix with respect to basis, using transition matricesTransformation Between Polynomials and Matrices Using Basis VectorsEuler representation of rotation matrix and its uniquenessProjective Transformation between two matched 3D point setsAre the transformation matrices the relation between basis $beta$ and $beta'$?
How to bypass password on Windows XP account?
Is there a problem creating Diff Backups every hour instead of Logs and DIffs?
How widely used is the term Treppenwitz? Is it something that most Germans know?
What to do with chalk when deepwater soloing?
What's the purpose of writing one's academic biography in the third person?
What's the meaning of 間時肆拾貳 at a car parking sign
Bete Noir -- no dairy
How to align text above triangle figure
What is the meaning of the new sigil in Game of Thrones Season 8 intro?
Why aren't air breathing engines used as small first stages
What does "fit" mean in this sentence?
3 doors, three guards, one stone
When do you get frequent flier miles - when you buy, or when you fly?
When a candle burns, why does the top of wick glow if bottom of flame is hottest?
Is the Standard Deduction better than Itemized when both are the same amount?
Dating a Former Employee
Should I discuss the type of campaign with my players?
Short Story with Cinderella as a Voo-doo Witch
At the end of Thor: Ragnarok why don't the Asgardians turn and head for the Bifrost as per their original plan?
If a contract sometimes uses the wrong name, is it still valid?
2001: A Space Odyssey's use of the song "Daisy Bell" (Bicycle Built for Two); life imitates art or vice-versa?
Echoing a tail command produces unexpected output?
Generate an RGB colour grid
What does an IRS interview request entail when called in to verify expenses for a sole proprietor small business?
Transforming back and forth between reference frames using orthogonal transformation matrices
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Q: Bases and Frames using Fourier SeriesQuestion about orthogonal transformation / orthogonal matricesHow can I interpret $2 times 2$ and $3 times 3$ transformation matrices geometrically?How to compute determinant (or eigenvalues) of this matrix?Properties of orthogonal, singular and antisymmetric matrices.Linear transformation matrix with respect to basis, using transition matricesTransformation Between Polynomials and Matrices Using Basis VectorsEuler representation of rotation matrix and its uniquenessProjective Transformation between two matched 3D point setsAre the transformation matrices the relation between basis $beta$ and $beta'$?
$begingroup$
The transformation of a covariance matrix $C$ from reference frame 1 to reference frame 2 is described as
beginequation
C_2 = R_12C_1R_12^T
endequation
using the (orthogonal) transformation matrix $R_12$, where the superscript $T$ denotes the transpose. In this case, I know what $C_2$ is and I want to find $C_1$. How do I use these transformation matrices in order to get $C_1$?
linear-algebra linear-transformations orthogonal-matrices
asked Mar 26 at 16:43
WouterWouter
31
$endgroup$
add a comment |
$begingroup$
The transformation of a covariance matrix $C$ from reference frame 1 to reference frame 2 is described as
beginequation
C_2 = R_12C_1R_12^T
endequation
using the (orthogonal) transformation matrix $R_12$, where the superscript $T$ denotes the transpose. In this case, I know what $C_2$ is and I want to find $C_1$. How do I use these transformation matrices in order to get $C_1$?
linear-algebra linear-transformations orthogonal-matrices
asked Mar 26 at 16:43
WouterWouter
31
$endgroup$
$begingroup$
Multiply by inverses on both sides and use the fact that for an orthogonal matrix $R^T=R^-1$.
$endgroup$
– amd
Mar 26 at 23:06
$begingroup$
So how does that look like mathematically? I am not sure on the notation... Would the equation then be $C_1$ = $R_12^TC_2R_12$?
$endgroup$
– Wouter
Mar 27 at 7:18
$begingroup$
Just so. I’ll write that up as an answer.
$endgroup$
– amd
Mar 27 at 17:49
add a comment |
$begingroup$
The transformation of a covariance matrix $C$ from reference frame 1 to reference frame 2 is described as
beginequation
C_2 = R_12C_1R_12^T
endequation
using the (orthogonal) transformation matrix $R_12$, where the superscript $T$ denotes the transpose. In this case, I know what $C_2$ is and I want to find $C_1$. How do I use these transformation matrices in order to get $C_1$?
linear-algebra linear-transformations orthogonal-matrices
asked Mar 26 at 16:43
WouterWouter
31
$endgroup$
The transformation of a covariance matrix $C$ from reference frame 1 to reference frame 2 is described as
beginequation
C_2 = R_12C_1R_12^T
endequation
using the (orthogonal) transformation matrix $R_12$, where the superscript $T$ denotes the transpose. In this case, I know what $C_2$ is and I want to find $C_1$. How do I use these transformation matrices in order to get $C_1$?
linear-algebra linear-transformations orthogonal-matrices
linear-algebra linear-transformations orthogonal-matrices
asked Mar 26 at 16:43
WouterWouter
31
asked Mar 26 at 16:43
WouterWouter
31
asked Mar 26 at 16:43
WouterWouter
31
asked Mar 26 at 16:43
WouterWouter
31
asked Mar 26 at 16:43
WouterWouter
31
31
$begingroup$
Multiply by inverses on both sides and use the fact that for an orthogonal matrix $R^T=R^-1$.
$endgroup$
– amd
Mar 26 at 23:06
$begingroup$
So how does that look like mathematically? I am not sure on the notation... Would the equation then be $C_1$ = $R_12^TC_2R_12$?
$endgroup$
– Wouter
Mar 27 at 7:18
$begingroup$
Just so. I’ll write that up as an answer.
$endgroup$
– amd
Mar 27 at 17:49
add a comment |
$begingroup$
Multiply by inverses on both sides and use the fact that for an orthogonal matrix $R^T=R^-1$.
$endgroup$
– amd
Mar 26 at 23:06
$begingroup$
So how does that look like mathematically? I am not sure on the notation... Would the equation then be $C_1$ = $R_12^TC_2R_12$?
$endgroup$
– Wouter
Mar 27 at 7:18
$begingroup$
Just so. I’ll write that up as an answer.
$endgroup$
– amd
Mar 27 at 17:49
$begingroup$
Multiply by inverses on both sides and use the fact that for an orthogonal matrix $R^T=R^-1$.
$endgroup$
– amd
Mar 26 at 23:06
$begingroup$
Multiply by inverses on both sides and use the fact that for an orthogonal matrix $R^T=R^-1$.
$endgroup$
– amd
Mar 26 at 23:06
$begingroup$
So how does that look like mathematically? I am not sure on the notation... Would the equation then be $C_1$ = $R_12^TC_2R_12$?
$endgroup$
– Wouter
Mar 27 at 7:18
$begingroup$
So how does that look like mathematically? I am not sure on the notation... Would the equation then be $C_1$ = $R_12^TC_2R_12$?
$endgroup$
– Wouter
Mar 27 at 7:18
$begingroup$
Just so. I’ll write that up as an answer.
$endgroup$
– amd
Mar 27 at 17:49
$begingroup$
Just so. I’ll write that up as an answer.
$endgroup$
– amd
Mar 27 at 17:49
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Isolate $C_1$ by multiplying by the appropriate inverses on both sides: $$C_1 = R_12^-1C_2R_12^-T$$ and then, since $R_12$ is orthogonal so that $R_12^-1=R_12^T$, $$C_1 = R_12^TC_2R_12.$$
answered Mar 27 at 17:52
amdamd
31.8k21053
$endgroup$
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%2f3163449%2ftransforming-back-and-forth-between-reference-frames-using-orthogonal-transforma%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
$begingroup$
Isolate $C_1$ by multiplying by the appropriate inverses on both sides: $$C_1 = R_12^-1C_2R_12^-T$$ and then, since $R_12$ is orthogonal so that $R_12^-1=R_12^T$, $$C_1 = R_12^TC_2R_12.$$
answered Mar 27 at 17:52
amdamd
31.8k21053
$endgroup$
add a comment |
$begingroup$
Isolate $C_1$ by multiplying by the appropriate inverses on both sides: $$C_1 = R_12^-1C_2R_12^-T$$ and then, since $R_12$ is orthogonal so that $R_12^-1=R_12^T$, $$C_1 = R_12^TC_2R_12.$$
answered Mar 27 at 17:52
amdamd
31.8k21053
$endgroup$
add a comment |
$begingroup$
Isolate $C_1$ by multiplying by the appropriate inverses on both sides: $$C_1 = R_12^-1C_2R_12^-T$$ and then, since $R_12$ is orthogonal so that $R_12^-1=R_12^T$, $$C_1 = R_12^TC_2R_12.$$
answered Mar 27 at 17:52
amdamd
31.8k21053
$endgroup$
Isolate $C_1$ by multiplying by the appropriate inverses on both sides: $$C_1 = R_12^-1C_2R_12^-T$$ and then, since $R_12$ is orthogonal so that $R_12^-1=R_12^T$, $$C_1 = R_12^TC_2R_12.$$
answered Mar 27 at 17:52
amdamd
31.8k21053
answered Mar 27 at 17:52
amdamd
31.8k21053
answered Mar 27 at 17:52
amdamd
31.8k21053
answered Mar 27 at 17:52
amdamd
31.8k21053
31.8k21053
add a comment |
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%2f3163449%2ftransforming-back-and-forth-between-reference-frames-using-orthogonal-transforma%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
$begingroup$
Multiply by inverses on both sides and use the fact that for an orthogonal matrix $R^T=R^-1$.
$endgroup$
– amd
Mar 26 at 23:06
$begingroup$
So how does that look like mathematically? I am not sure on the notation... Would the equation then be $C_1$ = $R_12^TC_2R_12$?
$endgroup$
– Wouter
Mar 27 at 7:18
$begingroup$
Just so. I’ll write that up as an answer.
$endgroup$
– amd
Mar 27 at 17:49