Finding the norm of $w + frac{1 - |w|^2}{|w - z|^2}(w - z)$, where $w$ and $z$ are in $mathbb{R}^n$
$begingroup$
I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function
$$M_w(z) = w + frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z in mathbb{R}^n$ and $|w| < 1$. (I am using different variable names than Stoll). This function generalizes the Möbius transformations on the complex numbers.
Stoll writes that
$$|M_w(z)|^2 = frac{|w - z|^2 + (1 - |w|^2)(1-|z|^2)}{|w - z|^2}$$
How do you show this? I tried computing $langle M_w(z), M_w(z) rangle$, but I'm not getting any closer.
This result ultimately shows that $M_w(z)$ maps into the unit ball.
hyperbolic-geometry mobius-transformation
$endgroup$
add a comment |
$begingroup$
I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function
$$M_w(z) = w + frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z in mathbb{R}^n$ and $|w| < 1$. (I am using different variable names than Stoll). This function generalizes the Möbius transformations on the complex numbers.
Stoll writes that
$$|M_w(z)|^2 = frac{|w - z|^2 + (1 - |w|^2)(1-|z|^2)}{|w - z|^2}$$
How do you show this? I tried computing $langle M_w(z), M_w(z) rangle$, but I'm not getting any closer.
This result ultimately shows that $M_w(z)$ maps into the unit ball.
hyperbolic-geometry mobius-transformation
$endgroup$
add a comment |
$begingroup$
I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function
$$M_w(z) = w + frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z in mathbb{R}^n$ and $|w| < 1$. (I am using different variable names than Stoll). This function generalizes the Möbius transformations on the complex numbers.
Stoll writes that
$$|M_w(z)|^2 = frac{|w - z|^2 + (1 - |w|^2)(1-|z|^2)}{|w - z|^2}$$
How do you show this? I tried computing $langle M_w(z), M_w(z) rangle$, but I'm not getting any closer.
This result ultimately shows that $M_w(z)$ maps into the unit ball.
hyperbolic-geometry mobius-transformation
$endgroup$
I'm trying to show a result from Manfred Stoll's book on Hyperbolic geometry. Consider the function
$$M_w(z) = w + frac{1 - |w|^2}{|w - z|^2}(w - z)$$ where $w,z in mathbb{R}^n$ and $|w| < 1$. (I am using different variable names than Stoll). This function generalizes the Möbius transformations on the complex numbers.
Stoll writes that
$$|M_w(z)|^2 = frac{|w - z|^2 + (1 - |w|^2)(1-|z|^2)}{|w - z|^2}$$
How do you show this? I tried computing $langle M_w(z), M_w(z) rangle$, but I'm not getting any closer.
This result ultimately shows that $M_w(z)$ maps into the unit ball.
hyperbolic-geometry mobius-transformation
hyperbolic-geometry mobius-transformation
edited Dec 3 '18 at 2:36
Blue
48k870153
48k870153
asked Dec 2 '18 at 21:47
dinstructiondinstruction
554423
554423
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
$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%2f3023257%2ffinding-the-norm-of-w-frac1-w2w-z2w-z-where-w-and-z%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$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
$endgroup$
add a comment |
$begingroup$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
$endgroup$
add a comment |
$begingroup$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
$endgroup$
$$begin{align}
left|M_w(z)right|^2 &=
|w|^2+frac{left(1-|w|^2right)^2}{|w-z|^4}|w-z|^2+2frac{1-|w|^2}{|w-z|^2};wcdot(w-z) \[8pt]
|w-z|^2;left|M_w(z)right|^2 &=
|w|^2|w-z|^2+left(1-|w|^2right)^2+2left(1-|w|^2right);left(|w|^2-wcdot zright) \[8pt]
&=
phantom{+2}|w|^2left(|w|^2+|z|^2-2wcdot zright)\
&phantom{=}+phantom{2}left(1-2|w|^2+|w|^4right)\
&phantom{=}+2left(|w|^2-wcdot z-|w|^4+|w|^2wcdot z right) \[8pt]
&=
left(-2wcdot zright)+ left(1 + |w|^2|z|^2right)\[6pt]
&=
left(|w|^2+|z|^2-2wcdot zright)+ left(1 -|w|^2-|z|^2+ |w|^2|z|^2right)\[6pt]
&=
|w-z|^2+ left(1 -|w|^2right)left(1-|z|^2right)\
end{align}$$
answered Dec 3 '18 at 2:33
BlueBlue
48k870153
48k870153
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%2f3023257%2ffinding-the-norm-of-w-frac1-w2w-z2w-z-where-w-and-z%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