Find the solution of an outer product induced system
Sorry if the question is lame, but I'm struggling to find the answer to the following problem:
Given a matrix $Ain mathbb{R}^{n,n}$ and a column vector $bin mathbb{R}^{n}$, how can one find the values of vectors $u in mathbb{R}^{n}$ such that
$$ b = Acirc u u^T mathbb{1}$$
where $circ$ is the elementwise (Hadamard) product, $mathbb{1}$ is a column vectors with ones and $x y^T$ is the outer product between $x$ and $y$. This means that for the $i_{th}$ element we can write:
$$b_i = sum_{j=1}^n a_{i,j} u_i u_j $$
The problems is quadratic, but the number of variables is equal to $n$, so I think there should be an easy solution to this class of problems.
Mybe the problem is known with some particular name, but I cannot find it in the literature. Hope you have some suggestions, thanks!
systems-of-equations inverse-problems quadrics outer-product
asked Nov 28 at 19:18
Lorenzo Nespoli
112
add a comment |
Sorry if the question is lame, but I'm struggling to find the answer to the following problem:
Given a matrix $Ain mathbb{R}^{n,n}$ and a column vector $bin mathbb{R}^{n}$, how can one find the values of vectors $u in mathbb{R}^{n}$ such that
$$ b = Acirc u u^T mathbb{1}$$
where $circ$ is the elementwise (Hadamard) product, $mathbb{1}$ is a column vectors with ones and $x y^T$ is the outer product between $x$ and $y$. This means that for the $i_{th}$ element we can write:
$$b_i = sum_{j=1}^n a_{i,j} u_i u_j $$
The problems is quadratic, but the number of variables is equal to $n$, so I think there should be an easy solution to this class of problems.
Mybe the problem is known with some particular name, but I cannot find it in the literature. Hope you have some suggestions, thanks!
systems-of-equations inverse-problems quadrics outer-product
asked Nov 28 at 19:18
Lorenzo Nespoli
112
add a comment |
Sorry if the question is lame, but I'm struggling to find the answer to the following problem:
Given a matrix $Ain mathbb{R}^{n,n}$ and a column vector $bin mathbb{R}^{n}$, how can one find the values of vectors $u in mathbb{R}^{n}$ such that
$$ b = Acirc u u^T mathbb{1}$$
where $circ$ is the elementwise (Hadamard) product, $mathbb{1}$ is a column vectors with ones and $x y^T$ is the outer product between $x$ and $y$. This means that for the $i_{th}$ element we can write:
$$b_i = sum_{j=1}^n a_{i,j} u_i u_j $$
The problems is quadratic, but the number of variables is equal to $n$, so I think there should be an easy solution to this class of problems.
Mybe the problem is known with some particular name, but I cannot find it in the literature. Hope you have some suggestions, thanks!
systems-of-equations inverse-problems quadrics outer-product
asked Nov 28 at 19:18
Lorenzo Nespoli
112
Sorry if the question is lame, but I'm struggling to find the answer to the following problem:
Given a matrix $Ain mathbb{R}^{n,n}$ and a column vector $bin mathbb{R}^{n}$, how can one find the values of vectors $u in mathbb{R}^{n}$ such that
$$ b = Acirc u u^T mathbb{1}$$
where $circ$ is the elementwise (Hadamard) product, $mathbb{1}$ is a column vectors with ones and $x y^T$ is the outer product between $x$ and $y$. This means that for the $i_{th}$ element we can write:
$$b_i = sum_{j=1}^n a_{i,j} u_i u_j $$
The problems is quadratic, but the number of variables is equal to $n$, so I think there should be an easy solution to this class of problems.
Mybe the problem is known with some particular name, but I cannot find it in the literature. Hope you have some suggestions, thanks!
systems-of-equations inverse-problems quadrics outer-product
systems-of-equations inverse-problems quadrics outer-product
asked Nov 28 at 19:18
Lorenzo Nespoli
112
asked Nov 28 at 19:18
Lorenzo Nespoli
112
edited Nov 28 at 19:24
asked Nov 28 at 19:18
Lorenzo Nespoli
112
asked Nov 28 at 19:18
Lorenzo Nespoli
112
asked Nov 28 at 19:18
Lorenzo Nespoli
112
112
add a comment |
add a comment |
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%2f3017570%2ffind-the-solution-of-an-outer-product-induced-system%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3017570%2ffind-the-solution-of-an-outer-product-induced-system%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
