Mixing
How to find the general linear transformation that preserves a certain scalar product?
$begingroup$
So I have got the scalar product of two vectors defined in this way:
$$(x,y)_c = x_1y_1+cx_2y_2$$ where $x = (x1,x2)$ and $y=(y1,y2)$. Now I need to find the set of tranformation matrices that preserves the scalar product defined above. I know that for the standard scalar product between two vectors the set of orthogonal matrices ${R_n}$ s.t. $R^T R = I$ preserves the scalar products:
$$(Rp, Rq) = (p,q) forall p,q$$
How do I find the equivalent transformation matrices for the scalar product above?
linear-algebra matrices linear-transformations
asked Jan 14 at 16:37
daljit97daljit97
178111
$endgroup$
add a comment |
$begingroup$
So I have got the scalar product of two vectors defined in this way:
$$(x,y)_c = x_1y_1+cx_2y_2$$ where $x = (x1,x2)$ and $y=(y1,y2)$. Now I need to find the set of tranformation matrices that preserves the scalar product defined above. I know that for the standard scalar product between two vectors the set of orthogonal matrices ${R_n}$ s.t. $R^T R = I$ preserves the scalar products:
$$(Rp, Rq) = (p,q) forall p,q$$
How do I find the equivalent transformation matrices for the scalar product above?
linear-algebra matrices linear-transformations
asked Jan 14 at 16:37
daljit97daljit97
178111
$endgroup$
add a comment |
$begingroup$
So I have got the scalar product of two vectors defined in this way:
$$(x,y)_c = x_1y_1+cx_2y_2$$ where $x = (x1,x2)$ and $y=(y1,y2)$. Now I need to find the set of tranformation matrices that preserves the scalar product defined above. I know that for the standard scalar product between two vectors the set of orthogonal matrices ${R_n}$ s.t. $R^T R = I$ preserves the scalar products:
$$(Rp, Rq) = (p,q) forall p,q$$
How do I find the equivalent transformation matrices for the scalar product above?
linear-algebra matrices linear-transformations
asked Jan 14 at 16:37
daljit97daljit97
178111
$endgroup$
So I have got the scalar product of two vectors defined in this way:
$$(x,y)_c = x_1y_1+cx_2y_2$$ where $x = (x1,x2)$ and $y=(y1,y2)$. Now I need to find the set of tranformation matrices that preserves the scalar product defined above. I know that for the standard scalar product between two vectors the set of orthogonal matrices ${R_n}$ s.t. $R^T R = I$ preserves the scalar products:
$$(Rp, Rq) = (p,q) forall p,q$$
How do I find the equivalent transformation matrices for the scalar product above?
linear-algebra matrices linear-transformations
linear-algebra matrices linear-transformations
asked Jan 14 at 16:37
daljit97daljit97
178111
asked Jan 14 at 16:37
daljit97daljit97
178111
edited Jan 14 at 16:53
daljit97
asked Jan 14 at 16:37
daljit97daljit97
178111
asked Jan 14 at 16:37
daljit97daljit97
178111
asked Jan 14 at 16:37
daljit97daljit97
178111
178111
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $C : c_{1,1} = 1, c_{2,2} = c, c_{1,2}=c_{2,1} = 0$. You want $(ACx)^TAy = (Cx)^Ty$, that is: $x^TCA^TAy = (Cx)^Ty$, then once again $A^TA=I$.
Edit:
Note that you can extend that to arbitrary diagonal matrices $C$. So you can literally have $x cdot_c y = sum_{i=0}^{n}{c_{i,i}x_iy_i}$ in arbitrary dimensions and the requirement will still be $A^TA=I$ as seen above.
answered Jan 14 at 17:19
lightxbulblightxbulb
900211
$endgroup$
$begingroup$
What do you mean by the notation $C : c_{1,1}$?
$endgroup$
– daljit97
Jan 14 at 18:19
$begingroup$
A matrix $C$, $c_{i,j}$ is the element in row $i$ and column $j$.
$endgroup$
– lightxbulb
Jan 14 at 18:28
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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%2f3073421%2fhow-to-find-the-general-linear-transformation-that-preserves-a-certain-scalar-pr%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
$begingroup$
Let $C : c_{1,1} = 1, c_{2,2} = c, c_{1,2}=c_{2,1} = 0$. You want $(ACx)^TAy = (Cx)^Ty$, that is: $x^TCA^TAy = (Cx)^Ty$, then once again $A^TA=I$.
Edit:
Note that you can extend that to arbitrary diagonal matrices $C$. So you can literally have $x cdot_c y = sum_{i=0}^{n}{c_{i,i}x_iy_i}$ in arbitrary dimensions and the requirement will still be $A^TA=I$ as seen above.
answered Jan 14 at 17:19
lightxbulblightxbulb
900211
$endgroup$
$begingroup$
What do you mean by the notation $C : c_{1,1}$?
$endgroup$
– daljit97
Jan 14 at 18:19
$begingroup$
A matrix $C$, $c_{i,j}$ is the element in row $i$ and column $j$.
$endgroup$
– lightxbulb
Jan 14 at 18:28
add a comment |
$begingroup$
Let $C : c_{1,1} = 1, c_{2,2} = c, c_{1,2}=c_{2,1} = 0$. You want $(ACx)^TAy = (Cx)^Ty$, that is: $x^TCA^TAy = (Cx)^Ty$, then once again $A^TA=I$.
Edit:
Note that you can extend that to arbitrary diagonal matrices $C$. So you can literally have $x cdot_c y = sum_{i=0}^{n}{c_{i,i}x_iy_i}$ in arbitrary dimensions and the requirement will still be $A^TA=I$ as seen above.
answered Jan 14 at 17:19
lightxbulblightxbulb
900211
$endgroup$
$begingroup$
What do you mean by the notation $C : c_{1,1}$?
$endgroup$
– daljit97
Jan 14 at 18:19
$begingroup$
A matrix $C$, $c_{i,j}$ is the element in row $i$ and column $j$.
$endgroup$
– lightxbulb
Jan 14 at 18:28
add a comment |
$begingroup$
Let $C : c_{1,1} = 1, c_{2,2} = c, c_{1,2}=c_{2,1} = 0$. You want $(ACx)^TAy = (Cx)^Ty$, that is: $x^TCA^TAy = (Cx)^Ty$, then once again $A^TA=I$.
Edit:
Note that you can extend that to arbitrary diagonal matrices $C$. So you can literally have $x cdot_c y = sum_{i=0}^{n}{c_{i,i}x_iy_i}$ in arbitrary dimensions and the requirement will still be $A^TA=I$ as seen above.
answered Jan 14 at 17:19
lightxbulblightxbulb
900211
$endgroup$
Let $C : c_{1,1} = 1, c_{2,2} = c, c_{1,2}=c_{2,1} = 0$. You want $(ACx)^TAy = (Cx)^Ty$, that is: $x^TCA^TAy = (Cx)^Ty$, then once again $A^TA=I$.
Edit:
Note that you can extend that to arbitrary diagonal matrices $C$. So you can literally have $x cdot_c y = sum_{i=0}^{n}{c_{i,i}x_iy_i}$ in arbitrary dimensions and the requirement will still be $A^TA=I$ as seen above.
answered Jan 14 at 17:19
lightxbulblightxbulb
900211
edited Jan 14 at 18:30
answered Jan 14 at 17:19
lightxbulblightxbulb
900211
answered Jan 14 at 17:19
lightxbulblightxbulb
900211
answered Jan 14 at 17:19
lightxbulblightxbulb
900211
900211
$begingroup$
What do you mean by the notation $C : c_{1,1}$?
$endgroup$
– daljit97
Jan 14 at 18:19
$begingroup$
A matrix $C$, $c_{i,j}$ is the element in row $i$ and column $j$.
$endgroup$
– lightxbulb
Jan 14 at 18:28
add a comment |
$begingroup$
What do you mean by the notation $C : c_{1,1}$?
$endgroup$
– daljit97
Jan 14 at 18:19
$begingroup$
A matrix $C$, $c_{i,j}$ is the element in row $i$ and column $j$.
$endgroup$
– lightxbulb
Jan 14 at 18:28
$begingroup$
What do you mean by the notation $C : c_{1,1}$?
$endgroup$
– daljit97
Jan 14 at 18:19
$begingroup$
What do you mean by the notation $C : c_{1,1}$?
$endgroup$
– daljit97
Jan 14 at 18:19
$begingroup$
A matrix $C$, $c_{i,j}$ is the element in row $i$ and column $j$.
$endgroup$
– lightxbulb
Jan 14 at 18:28
$begingroup$
A matrix $C$, $c_{i,j}$ is the element in row $i$ and column $j$.
$endgroup$
– lightxbulb
Jan 14 at 18:28
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%2f3073421%2fhow-to-find-the-general-linear-transformation-that-preserves-a-certain-scalar-pr%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
