Least Squares Solution of Minimal Norm when $A^{*}b = 0$
Suppose, given a matrix $textbf{A} in mathbb{C}^{m times n}$ and a vector $textbf{b} in mathbb{C}^{n}$, I want to find the minimal norm solution of
$$min_{textbf{x}}|textbf{A}textbf{x} - textbf{b} |_{2} $$
with the condition that $textbf{A}^{*}textbf{b} = textbf{0}_{n}$.
If $rank(textbf{A}) =n$, then the solution is
$$textbf{y} = (textbf{A}^{*}textbf{A})^{-1}textbf{A}^{*}textbf{b}.$$
This would give $textbf{y} = textbf{0}$ with the stated condition. However, for $textbf{A}$ of arbitrary rank, the the minimal norm solution for this problem is
$$textbf{y} = textbf{A}^{dagger}textbf{b}$$
Where $textbf{A}^{dagger}$ is the pseudoinverse.
Can we say anything about $textbf{y}$ with stated condition in the general case, or is it arbitrary
linear-algebra matrices numerical-linear-algebra least-squares constraints
asked Nov 28 '18 at 12:16
SimonP
434
add a comment |
Suppose, given a matrix $textbf{A} in mathbb{C}^{m times n}$ and a vector $textbf{b} in mathbb{C}^{n}$, I want to find the minimal norm solution of
$$min_{textbf{x}}|textbf{A}textbf{x} - textbf{b} |_{2} $$
with the condition that $textbf{A}^{*}textbf{b} = textbf{0}_{n}$.
If $rank(textbf{A}) =n$, then the solution is
$$textbf{y} = (textbf{A}^{*}textbf{A})^{-1}textbf{A}^{*}textbf{b}.$$
This would give $textbf{y} = textbf{0}$ with the stated condition. However, for $textbf{A}$ of arbitrary rank, the the minimal norm solution for this problem is
$$textbf{y} = textbf{A}^{dagger}textbf{b}$$
Where $textbf{A}^{dagger}$ is the pseudoinverse.
Can we say anything about $textbf{y}$ with stated condition in the general case, or is it arbitrary
linear-algebra matrices numerical-linear-algebra least-squares constraints
asked Nov 28 '18 at 12:16
SimonP
434
add a comment |
Suppose, given a matrix $textbf{A} in mathbb{C}^{m times n}$ and a vector $textbf{b} in mathbb{C}^{n}$, I want to find the minimal norm solution of
$$min_{textbf{x}}|textbf{A}textbf{x} - textbf{b} |_{2} $$
with the condition that $textbf{A}^{*}textbf{b} = textbf{0}_{n}$.
If $rank(textbf{A}) =n$, then the solution is
$$textbf{y} = (textbf{A}^{*}textbf{A})^{-1}textbf{A}^{*}textbf{b}.$$
This would give $textbf{y} = textbf{0}$ with the stated condition. However, for $textbf{A}$ of arbitrary rank, the the minimal norm solution for this problem is
$$textbf{y} = textbf{A}^{dagger}textbf{b}$$
Where $textbf{A}^{dagger}$ is the pseudoinverse.
Can we say anything about $textbf{y}$ with stated condition in the general case, or is it arbitrary
linear-algebra matrices numerical-linear-algebra least-squares constraints
asked Nov 28 '18 at 12:16
SimonP
434
Suppose, given a matrix $textbf{A} in mathbb{C}^{m times n}$ and a vector $textbf{b} in mathbb{C}^{n}$, I want to find the minimal norm solution of
$$min_{textbf{x}}|textbf{A}textbf{x} - textbf{b} |_{2} $$
with the condition that $textbf{A}^{*}textbf{b} = textbf{0}_{n}$.
If $rank(textbf{A}) =n$, then the solution is
$$textbf{y} = (textbf{A}^{*}textbf{A})^{-1}textbf{A}^{*}textbf{b}.$$
This would give $textbf{y} = textbf{0}$ with the stated condition. However, for $textbf{A}$ of arbitrary rank, the the minimal norm solution for this problem is
$$textbf{y} = textbf{A}^{dagger}textbf{b}$$
Where $textbf{A}^{dagger}$ is the pseudoinverse.
Can we say anything about $textbf{y}$ with stated condition in the general case, or is it arbitrary
linear-algebra matrices numerical-linear-algebra least-squares constraints
linear-algebra matrices numerical-linear-algebra least-squares constraints
asked Nov 28 '18 at 12:16
SimonP
434
asked Nov 28 '18 at 12:16
SimonP
434
asked Nov 28 '18 at 12:16
SimonP
434
asked Nov 28 '18 at 12:16
SimonP
434
asked Nov 28 '18 at 12:16
SimonP
434
434
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
It is known that a vector $hat {mathbf x}$ is a least squares solution of the system
$$
A {mathbf x}= {mathbf b}
$$ iff $$A^*Ahat {mathbf x}=A^* {mathbf b}.$$
Thus, if the condition $A^* {mathbf b}={mathbf 0}$ is satisfied, then $hat {mathbf x}= {mathbf 0}$ is a least squares solution. Obviously, it has the minimum possible norm (zero).
answered Nov 28 '18 at 15:14
AVK
2,0711517
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%2f3017078%2fleast-squares-solution-of-minimal-norm-when-ab-0%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
It is known that a vector $hat {mathbf x}$ is a least squares solution of the system
$$
A {mathbf x}= {mathbf b}
$$ iff $$A^*Ahat {mathbf x}=A^* {mathbf b}.$$
Thus, if the condition $A^* {mathbf b}={mathbf 0}$ is satisfied, then $hat {mathbf x}= {mathbf 0}$ is a least squares solution. Obviously, it has the minimum possible norm (zero).
answered Nov 28 '18 at 15:14
AVK
2,0711517
add a comment |
It is known that a vector $hat {mathbf x}$ is a least squares solution of the system
$$
A {mathbf x}= {mathbf b}
$$ iff $$A^*Ahat {mathbf x}=A^* {mathbf b}.$$
Thus, if the condition $A^* {mathbf b}={mathbf 0}$ is satisfied, then $hat {mathbf x}= {mathbf 0}$ is a least squares solution. Obviously, it has the minimum possible norm (zero).
answered Nov 28 '18 at 15:14
AVK
2,0711517
add a comment |
It is known that a vector $hat {mathbf x}$ is a least squares solution of the system
$$
A {mathbf x}= {mathbf b}
$$ iff $$A^*Ahat {mathbf x}=A^* {mathbf b}.$$
Thus, if the condition $A^* {mathbf b}={mathbf 0}$ is satisfied, then $hat {mathbf x}= {mathbf 0}$ is a least squares solution. Obviously, it has the minimum possible norm (zero).
answered Nov 28 '18 at 15:14
AVK
2,0711517
It is known that a vector $hat {mathbf x}$ is a least squares solution of the system
$$
A {mathbf x}= {mathbf b}
$$ iff $$A^*Ahat {mathbf x}=A^* {mathbf b}.$$
Thus, if the condition $A^* {mathbf b}={mathbf 0}$ is satisfied, then $hat {mathbf x}= {mathbf 0}$ is a least squares solution. Obviously, it has the minimum possible norm (zero).
answered Nov 28 '18 at 15:14
AVK
2,0711517
answered Nov 28 '18 at 15:14
AVK
2,0711517
answered Nov 28 '18 at 15:14
AVK
2,0711517
answered Nov 28 '18 at 15:14
AVK
2,0711517
2,0711517
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.
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%2f3017078%2fleast-squares-solution-of-minimal-norm-when-ab-0%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
