Isotropic points with a jacobian matrix
$begingroup$
As definition, in order to find the isotropic points of a jacobian matrix, the matrix's columns become orthogonal and equal to the magnitude. I don't quite understand this definition. if i have a matrix looks like this:
a b
c d
Matrix example:
2.x 0
2.y+3 3
How can I find its isotropic points ?
Thank you very much.
matrix-calculus
asked Jan 21 '14 at 14:31
XitrumXitrum
1012
$endgroup$
add a comment |
$begingroup$
As definition, in order to find the isotropic points of a jacobian matrix, the matrix's columns become orthogonal and equal to the magnitude. I don't quite understand this definition. if i have a matrix looks like this:
a b
c d
Matrix example:
2.x 0
2.y+3 3
How can I find its isotropic points ?
Thank you very much.
matrix-calculus
asked Jan 21 '14 at 14:31
XitrumXitrum
1012
$endgroup$
add a comment |
$begingroup$
As definition, in order to find the isotropic points of a jacobian matrix, the matrix's columns become orthogonal and equal to the magnitude. I don't quite understand this definition. if i have a matrix looks like this:
a b
c d
Matrix example:
2.x 0
2.y+3 3
How can I find its isotropic points ?
Thank you very much.
matrix-calculus
asked Jan 21 '14 at 14:31
XitrumXitrum
1012
$endgroup$
As definition, in order to find the isotropic points of a jacobian matrix, the matrix's columns become orthogonal and equal to the magnitude. I don't quite understand this definition. if i have a matrix looks like this:
a b
c d
Matrix example:
2.x 0
2.y+3 3
How can I find its isotropic points ?
Thank you very much.
matrix-calculus
matrix-calculus
asked Jan 21 '14 at 14:31
XitrumXitrum
1012
asked Jan 21 '14 at 14:31
XitrumXitrum
1012
asked Jan 21 '14 at 14:31
XitrumXitrum
1012
asked Jan 21 '14 at 14:31
XitrumXitrum
1012
asked Jan 21 '14 at 14:31
XitrumXitrum
1012
1012
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The columns must be orthogonal to each other (i.e., the dot product of any two columns must be equal to $0$), and their vector norms must be equal. For the general $2times 2$ matrix
$$
begin{pmatrix}
a & b \
c & d
end{pmatrix}
$$
these requirements amount to: $ab+cd=0$, $a^2+c^2=b^2+d^2$.
For the concrete example
$$
begin{pmatrix}
2x & 0 \
2y+3 & 3
end{pmatrix}
$$
the above requirements simplify to $2y+3=0$; $(2x)^2+(2y+3)^2 = 9$. Hence $y=-3/2$ and $x=pm 3/2$.
answered Mar 2 '14 at 0:22
user127096user127096
8,17011140
$endgroup$
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%2f646240%2fisotropic-points-with-a-jacobian-matrix%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
$begingroup$
The columns must be orthogonal to each other (i.e., the dot product of any two columns must be equal to $0$), and their vector norms must be equal. For the general $2times 2$ matrix
$$
begin{pmatrix}
a & b \
c & d
end{pmatrix}
$$
these requirements amount to: $ab+cd=0$, $a^2+c^2=b^2+d^2$.
For the concrete example
$$
begin{pmatrix}
2x & 0 \
2y+3 & 3
end{pmatrix}
$$
the above requirements simplify to $2y+3=0$; $(2x)^2+(2y+3)^2 = 9$. Hence $y=-3/2$ and $x=pm 3/2$.
answered Mar 2 '14 at 0:22
user127096user127096
8,17011140
$endgroup$
add a comment |
$begingroup$
The columns must be orthogonal to each other (i.e., the dot product of any two columns must be equal to $0$), and their vector norms must be equal. For the general $2times 2$ matrix
$$
begin{pmatrix}
a & b \
c & d
end{pmatrix}
$$
these requirements amount to: $ab+cd=0$, $a^2+c^2=b^2+d^2$.
For the concrete example
$$
begin{pmatrix}
2x & 0 \
2y+3 & 3
end{pmatrix}
$$
the above requirements simplify to $2y+3=0$; $(2x)^2+(2y+3)^2 = 9$. Hence $y=-3/2$ and $x=pm 3/2$.
answered Mar 2 '14 at 0:22
user127096user127096
8,17011140
$endgroup$
add a comment |
$begingroup$
The columns must be orthogonal to each other (i.e., the dot product of any two columns must be equal to $0$), and their vector norms must be equal. For the general $2times 2$ matrix
$$
begin{pmatrix}
a & b \
c & d
end{pmatrix}
$$
these requirements amount to: $ab+cd=0$, $a^2+c^2=b^2+d^2$.
For the concrete example
$$
begin{pmatrix}
2x & 0 \
2y+3 & 3
end{pmatrix}
$$
the above requirements simplify to $2y+3=0$; $(2x)^2+(2y+3)^2 = 9$. Hence $y=-3/2$ and $x=pm 3/2$.
answered Mar 2 '14 at 0:22
user127096user127096
8,17011140
$endgroup$
The columns must be orthogonal to each other (i.e., the dot product of any two columns must be equal to $0$), and their vector norms must be equal. For the general $2times 2$ matrix
$$
begin{pmatrix}
a & b \
c & d
end{pmatrix}
$$
these requirements amount to: $ab+cd=0$, $a^2+c^2=b^2+d^2$.
For the concrete example
$$
begin{pmatrix}
2x & 0 \
2y+3 & 3
end{pmatrix}
$$
the above requirements simplify to $2y+3=0$; $(2x)^2+(2y+3)^2 = 9$. Hence $y=-3/2$ and $x=pm 3/2$.
answered Mar 2 '14 at 0:22
user127096user127096
8,17011140
answered Mar 2 '14 at 0:22
user127096user127096
8,17011140
answered Mar 2 '14 at 0:22
user127096user127096
8,17011140
answered Mar 2 '14 at 0:22
user127096user127096
8,17011140
8,17011140
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%2f646240%2fisotropic-points-with-a-jacobian-matrix%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
