Mixing
solving a linear matrix equation where the variable appears in a diagonal matrix
$begingroup$
For constant $m times 1$ vectors $c$ and $d$, constant $m times n$ matrix $Z$, and variable $n times 1$ vector $x$, I want to solve the following for $x$:
$$left[left(d^{T}diag(Zx)-c^{T}right)Zright]^{T}=0$$
I have searched for answers to similar questions on this site, and I am guessing that the answer may involve vectorization and the Kronecker product, but I think that the diagonalization operator is causing me trouble.
linear-algebra
edited Jan 31 at 19:26
Bara
7010
asked Jan 31 at 18:21
GrantGrant
63
$endgroup$
add a comment |
$begingroup$
For constant $m times 1$ vectors $c$ and $d$, constant $m times n$ matrix $Z$, and variable $n times 1$ vector $x$, I want to solve the following for $x$:
$$left[left(d^{T}diag(Zx)-c^{T}right)Zright]^{T}=0$$
I have searched for answers to similar questions on this site, and I am guessing that the answer may involve vectorization and the Kronecker product, but I think that the diagonalization operator is causing me trouble.
linear-algebra
edited Jan 31 at 19:26
Bara
7010
asked Jan 31 at 18:21
GrantGrant
63
$endgroup$
add a comment |
$begingroup$
For constant $m times 1$ vectors $c$ and $d$, constant $m times n$ matrix $Z$, and variable $n times 1$ vector $x$, I want to solve the following for $x$:
$$left[left(d^{T}diag(Zx)-c^{T}right)Zright]^{T}=0$$
I have searched for answers to similar questions on this site, and I am guessing that the answer may involve vectorization and the Kronecker product, but I think that the diagonalization operator is causing me trouble.
linear-algebra
edited Jan 31 at 19:26
Bara
7010
asked Jan 31 at 18:21
GrantGrant
63
$endgroup$
For constant $m times 1$ vectors $c$ and $d$, constant $m times n$ matrix $Z$, and variable $n times 1$ vector $x$, I want to solve the following for $x$:
$$left[left(d^{T}diag(Zx)-c^{T}right)Zright]^{T}=0$$
I have searched for answers to similar questions on this site, and I am guessing that the answer may involve vectorization and the Kronecker product, but I think that the diagonalization operator is causing me trouble.
linear-algebra
linear-algebra
edited Jan 31 at 19:26
Bara
7010
asked Jan 31 at 18:21
GrantGrant
63
edited Jan 31 at 19:26
Bara
7010
asked Jan 31 at 18:21
GrantGrant
63
edited Jan 31 at 19:26
Bara
7010
edited Jan 31 at 19:26
Bara
7010
edited Jan 31 at 19:26
Bara
7010
7010
asked Jan 31 at 18:21
GrantGrant
63
asked Jan 31 at 18:21
GrantGrant
63
asked Jan 31 at 18:21
GrantGrant
63
63
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I figured it out myself.
Form an $m times n$ matrix $P$, each column $j$ of which is equal to $-diag(Z^{<j>}d)$, where $Z^{<j>}$ denotes the $j$th column of $Z$. The solution is then given by $-(P^{T}Z)^{-1}Z^{T}c$.
answered Feb 12 at 21:27
GrantGrant
63
$endgroup$
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%2f3095267%2fsolving-a-linear-matrix-equation-where-the-variable-appears-in-a-diagonal-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
$begingroup$
I figured it out myself.
Form an $m times n$ matrix $P$, each column $j$ of which is equal to $-diag(Z^{<j>}d)$, where $Z^{<j>}$ denotes the $j$th column of $Z$. The solution is then given by $-(P^{T}Z)^{-1}Z^{T}c$.
answered Feb 12 at 21:27
GrantGrant
63
$endgroup$
add a comment |
$begingroup$
I figured it out myself.
Form an $m times n$ matrix $P$, each column $j$ of which is equal to $-diag(Z^{<j>}d)$, where $Z^{<j>}$ denotes the $j$th column of $Z$. The solution is then given by $-(P^{T}Z)^{-1}Z^{T}c$.
answered Feb 12 at 21:27
GrantGrant
63
$endgroup$
add a comment |
$begingroup$
I figured it out myself.
Form an $m times n$ matrix $P$, each column $j$ of which is equal to $-diag(Z^{<j>}d)$, where $Z^{<j>}$ denotes the $j$th column of $Z$. The solution is then given by $-(P^{T}Z)^{-1}Z^{T}c$.
answered Feb 12 at 21:27
GrantGrant
63
$endgroup$
I figured it out myself.
Form an $m times n$ matrix $P$, each column $j$ of which is equal to $-diag(Z^{<j>}d)$, where $Z^{<j>}$ denotes the $j$th column of $Z$. The solution is then given by $-(P^{T}Z)^{-1}Z^{T}c$.
answered Feb 12 at 21:27
GrantGrant
63
answered Feb 12 at 21:27
GrantGrant
63
answered Feb 12 at 21:27
GrantGrant
63
answered Feb 12 at 21:27
GrantGrant
63
63
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%2f3095267%2fsolving-a-linear-matrix-equation-where-the-variable-appears-in-a-diagonal-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
