Mixing
Induced inner product on tensor powers.
$begingroup$
Let $V$ be a real or complex inner product space with inner product $leftlangle cdotp,cdotprightrangle$. For $otimes ^kV$ define $leftlangle cdotp,cdotprightrangle_k$ by
$$leftlangle u_1otimes ...otimes u_k,v_1otimes ...otimes v_krightrangle_k = leftlangle u_1,v_1rightrangle...leftlangle u_k,v_krightrangle$$
on pure tensors, and extend bilinearly.
Is the resulting object $leftlangle cdotp,cdotprightrangle_k$ an inner product on $otimes ^kV$? It is clearly a symmetric bilinear form. Is it non-degenerate and positive definite though?
linear-algebra inner-product-space tensor-products multilinear-algebra
asked Jan 22 at 15:23
Joshua TilleyJoshua Tilley
556313
$endgroup$
add a comment |
$begingroup$
Let $V$ be a real or complex inner product space with inner product $leftlangle cdotp,cdotprightrangle$. For $otimes ^kV$ define $leftlangle cdotp,cdotprightrangle_k$ by
$$leftlangle u_1otimes ...otimes u_k,v_1otimes ...otimes v_krightrangle_k = leftlangle u_1,v_1rightrangle...leftlangle u_k,v_krightrangle$$
on pure tensors, and extend bilinearly.
Is the resulting object $leftlangle cdotp,cdotprightrangle_k$ an inner product on $otimes ^kV$? It is clearly a symmetric bilinear form. Is it non-degenerate and positive definite though?
linear-algebra inner-product-space tensor-products multilinear-algebra
asked Jan 22 at 15:23
Joshua TilleyJoshua Tilley
556313
$endgroup$
$begingroup$
do $v_i=u_i$ for all the $i$ and infer
$endgroup$
– janmarqz
Jan 26 at 17:35
$begingroup$
That only gives the result for tensors of the particular form above.
$endgroup$
– Joshua Tilley
Jan 29 at 19:25
$begingroup$
you need to do that in order to check your last question
$endgroup$
– janmarqz
Jan 30 at 3:23
add a comment |
$begingroup$
Let $V$ be a real or complex inner product space with inner product $leftlangle cdotp,cdotprightrangle$. For $otimes ^kV$ define $leftlangle cdotp,cdotprightrangle_k$ by
$$leftlangle u_1otimes ...otimes u_k,v_1otimes ...otimes v_krightrangle_k = leftlangle u_1,v_1rightrangle...leftlangle u_k,v_krightrangle$$
on pure tensors, and extend bilinearly.
Is the resulting object $leftlangle cdotp,cdotprightrangle_k$ an inner product on $otimes ^kV$? It is clearly a symmetric bilinear form. Is it non-degenerate and positive definite though?
linear-algebra inner-product-space tensor-products multilinear-algebra
asked Jan 22 at 15:23
Joshua TilleyJoshua Tilley
556313
$endgroup$
Let $V$ be a real or complex inner product space with inner product $leftlangle cdotp,cdotprightrangle$. For $otimes ^kV$ define $leftlangle cdotp,cdotprightrangle_k$ by
$$leftlangle u_1otimes ...otimes u_k,v_1otimes ...otimes v_krightrangle_k = leftlangle u_1,v_1rightrangle...leftlangle u_k,v_krightrangle$$
on pure tensors, and extend bilinearly.
Is the resulting object $leftlangle cdotp,cdotprightrangle_k$ an inner product on $otimes ^kV$? It is clearly a symmetric bilinear form. Is it non-degenerate and positive definite though?
linear-algebra inner-product-space tensor-products multilinear-algebra
linear-algebra inner-product-space tensor-products multilinear-algebra
asked Jan 22 at 15:23
Joshua TilleyJoshua Tilley
556313
asked Jan 22 at 15:23
Joshua TilleyJoshua Tilley
556313
asked Jan 22 at 15:23
Joshua TilleyJoshua Tilley
556313
asked Jan 22 at 15:23
Joshua TilleyJoshua Tilley
556313
asked Jan 22 at 15:23
Joshua TilleyJoshua Tilley
556313
556313
$begingroup$
do $v_i=u_i$ for all the $i$ and infer
$endgroup$
– janmarqz
Jan 26 at 17:35
$begingroup$
That only gives the result for tensors of the particular form above.
$endgroup$
– Joshua Tilley
Jan 29 at 19:25
$begingroup$
you need to do that in order to check your last question
$endgroup$
– janmarqz
Jan 30 at 3:23
add a comment |
$begingroup$
do $v_i=u_i$ for all the $i$ and infer
$endgroup$
– janmarqz
Jan 26 at 17:35
$begingroup$
That only gives the result for tensors of the particular form above.
$endgroup$
– Joshua Tilley
Jan 29 at 19:25
$begingroup$
you need to do that in order to check your last question
$endgroup$
– janmarqz
Jan 30 at 3:23
$begingroup$
do $v_i=u_i$ for all the $i$ and infer
$endgroup$
– janmarqz
Jan 26 at 17:35
$begingroup$
do $v_i=u_i$ for all the $i$ and infer
$endgroup$
– janmarqz
Jan 26 at 17:35
$begingroup$
That only gives the result for tensors of the particular form above.
$endgroup$
– Joshua Tilley
Jan 29 at 19:25
$begingroup$
That only gives the result for tensors of the particular form above.
$endgroup$
– Joshua Tilley
Jan 29 at 19:25
$begingroup$
you need to do that in order to check your last question
$endgroup$
– janmarqz
Jan 30 at 3:23
$begingroup$
you need to do that in order to check your last question
$endgroup$
– janmarqz
Jan 30 at 3:23
add a comment |
0
active
oldest
votes
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%2f3083295%2finduced-inner-product-on-tensor-powers%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3083295%2finduced-inner-product-on-tensor-powers%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
$begingroup$
do $v_i=u_i$ for all the $i$ and infer
$endgroup$
– janmarqz
Jan 26 at 17:35
$begingroup$
That only gives the result for tensors of the particular form above.
$endgroup$
– Joshua Tilley
Jan 29 at 19:25
$begingroup$
you need to do that in order to check your last question
$endgroup$
– janmarqz
Jan 30 at 3:23