Condition for a symmetric matrix to contain only positive entries?
I have a matrix defined as $mathbf{X} -mathbf{Y}mathbf{Y}^top$, $mathbf{Y}$ is a matrix containing only non-negative entries and $mathbf{X}$ is a similarity matrix, hence $mathbf{X}$ also contains only non-negative entries. The matrix $mathbf{X}$ is a PSD matrix and $mathbf{X} -mathbf{Y}mathbf{Y}^top$ is a symmetric matrix.
Now, is there any condition (other than the trivial condition that element-wise $mathbf{X} geq mathbf{YY}^top$).
I am looking for any condition in terms of the spectral properties of the matrices or the bound on the supremum of elements of the matrices etc.
linear-algebra matrices matrix-decomposition positive-semidefinite
asked Nov 27 '18 at 9:50
Shew
563413
add a comment |
I have a matrix defined as $mathbf{X} -mathbf{Y}mathbf{Y}^top$, $mathbf{Y}$ is a matrix containing only non-negative entries and $mathbf{X}$ is a similarity matrix, hence $mathbf{X}$ also contains only non-negative entries. The matrix $mathbf{X}$ is a PSD matrix and $mathbf{X} -mathbf{Y}mathbf{Y}^top$ is a symmetric matrix.
Now, is there any condition (other than the trivial condition that element-wise $mathbf{X} geq mathbf{YY}^top$).
I am looking for any condition in terms of the spectral properties of the matrices or the bound on the supremum of elements of the matrices etc.
linear-algebra matrices matrix-decomposition positive-semidefinite
asked Nov 27 '18 at 9:50
Shew
563413
1
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
– Eric
Nov 27 '18 at 10:34
add a comment |
I have a matrix defined as $mathbf{X} -mathbf{Y}mathbf{Y}^top$, $mathbf{Y}$ is a matrix containing only non-negative entries and $mathbf{X}$ is a similarity matrix, hence $mathbf{X}$ also contains only non-negative entries. The matrix $mathbf{X}$ is a PSD matrix and $mathbf{X} -mathbf{Y}mathbf{Y}^top$ is a symmetric matrix.
Now, is there any condition (other than the trivial condition that element-wise $mathbf{X} geq mathbf{YY}^top$).
I am looking for any condition in terms of the spectral properties of the matrices or the bound on the supremum of elements of the matrices etc.
linear-algebra matrices matrix-decomposition positive-semidefinite
asked Nov 27 '18 at 9:50
Shew
563413
I have a matrix defined as $mathbf{X} -mathbf{Y}mathbf{Y}^top$, $mathbf{Y}$ is a matrix containing only non-negative entries and $mathbf{X}$ is a similarity matrix, hence $mathbf{X}$ also contains only non-negative entries. The matrix $mathbf{X}$ is a PSD matrix and $mathbf{X} -mathbf{Y}mathbf{Y}^top$ is a symmetric matrix.
Now, is there any condition (other than the trivial condition that element-wise $mathbf{X} geq mathbf{YY}^top$).
I am looking for any condition in terms of the spectral properties of the matrices or the bound on the supremum of elements of the matrices etc.
linear-algebra matrices matrix-decomposition positive-semidefinite
linear-algebra matrices matrix-decomposition positive-semidefinite
asked Nov 27 '18 at 9:50
Shew
563413
asked Nov 27 '18 at 9:50
Shew
563413
asked Nov 27 '18 at 9:50
Shew
563413
asked Nov 27 '18 at 9:50
Shew
563413
asked Nov 27 '18 at 9:50
Shew
563413
563413
1
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
– Eric
Nov 27 '18 at 10:34
add a comment |
1
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
– Eric
Nov 27 '18 at 10:34
1
1
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
– Eric
Nov 27 '18 at 10:34
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
– Eric
Nov 27 '18 at 10:34
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%2f3015576%2fcondition-for-a-symmetric-matrix-to-contain-only-positive-entries%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.
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%2f3015576%2fcondition-for-a-symmetric-matrix-to-contain-only-positive-entries%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
1
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
– Eric
Nov 27 '18 at 10:34