A Matrix Inequality for positive definite matrices The Next CEO of Stack OverflowArithmetic-geometric mean of positive matricesEigenvalue distribution of positive-definite analytic functionDecomposition of positive definite matrices.A spectral inequality for positive-definite matrices determinant inequality for symmetric positive definite matricesDeterminant inequality involving Hermitian, positive definite matricesEigenvalues of a partitioned self-adjoint matrixA log inequality for positive definite trace-one matricesExpectation of square root of positive definite matrixProve that matrix is positive definite
A Matrix Inequality for positive definite matrices
The Next CEO of Stack OverflowArithmetic-geometric mean of positive matricesEigenvalue distribution of positive-definite analytic functionDecomposition of positive definite matrices.A spectral inequality for positive-definite matrices determinant inequality for symmetric positive definite matricesDeterminant inequality involving Hermitian, positive definite matricesEigenvalues of a partitioned self-adjoint matrixA log inequality for positive definite trace-one matricesExpectation of square root of positive definite matrixProve that matrix is positive definite
$begingroup$
Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^*A)^frac12$ ?
PS. I think the answer is No. But I could not find any counterexample!
linear-algebra matrices operator-theory
edited 2 mins ago
Denis Serre
29.6k795197
asked 4 hours ago
Samya Kumar RaySamya Kumar Ray
364
$endgroup$
add a comment |
$begingroup$
Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^*A)^frac12$ ?
PS. I think the answer is No. But I could not find any counterexample!
linear-algebra matrices operator-theory
edited 2 mins ago
Denis Serre
29.6k795197
asked 4 hours ago
Samya Kumar RaySamya Kumar Ray
364
$endgroup$
1
$begingroup$
Consider $X,Y$ to be of trace 1 and their difference to be block diagonal $sigma^x,sigma^x$. Now $|X-Y|=$ identity matrix. Take the trace of both sides, the LHS gives 4 the RHS is 2.
$endgroup$
– lcv
3 hours ago
$begingroup$
@lcv. can you please elaborate your notations?
$endgroup$
– Samya Kumar Ray
2 hours ago
add a comment |
$begingroup$
Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^*A)^frac12$ ?
PS. I think the answer is No. But I could not find any counterexample!
linear-algebra matrices operator-theory
edited 2 mins ago
Denis Serre
29.6k795197
asked 4 hours ago
Samya Kumar RaySamya Kumar Ray
364
$endgroup$
Let $X$ and $Y$ be positive semi-definite self-adjoint complex matrices of same finite order. The, is it true that $|X-Y|leq X+Y$ where for any matrix $A$, $|A|$ is defined to be $|A|:=(A^*A)^frac12$ ?
PS. I think the answer is No. But I could not find any counterexample!
linear-algebra matrices operator-theory
linear-algebra matrices operator-theory
edited 2 mins ago
Denis Serre
29.6k795197
asked 4 hours ago
Samya Kumar RaySamya Kumar Ray
364
edited 2 mins ago
Denis Serre
29.6k795197
asked 4 hours ago
Samya Kumar RaySamya Kumar Ray
364
edited 2 mins ago
Denis Serre
29.6k795197
edited 2 mins ago
Denis Serre
29.6k795197
edited 2 mins ago
Denis Serre
29.6k795197
29.6k795197
asked 4 hours ago
Samya Kumar RaySamya Kumar Ray
364
asked 4 hours ago
Samya Kumar RaySamya Kumar Ray
364
asked 4 hours ago
Samya Kumar RaySamya Kumar Ray
364
364
1
$begingroup$
Consider $X,Y$ to be of trace 1 and their difference to be block diagonal $sigma^x,sigma^x$. Now $|X-Y|=$ identity matrix. Take the trace of both sides, the LHS gives 4 the RHS is 2.
$endgroup$
– lcv
3 hours ago
$begingroup$
@lcv. can you please elaborate your notations?
$endgroup$
– Samya Kumar Ray
2 hours ago
add a comment |
1
$begingroup$
Consider $X,Y$ to be of trace 1 and their difference to be block diagonal $sigma^x,sigma^x$. Now $|X-Y|=$ identity matrix. Take the trace of both sides, the LHS gives 4 the RHS is 2.
$endgroup$
– lcv
3 hours ago
$begingroup$
@lcv. can you please elaborate your notations?
$endgroup$
– Samya Kumar Ray
2 hours ago
1
1
$begingroup$
Consider $X,Y$ to be of trace 1 and their difference to be block diagonal $sigma^x,sigma^x$. Now $|X-Y|=$ identity matrix. Take the trace of both sides, the LHS gives 4 the RHS is 2.
$endgroup$
– lcv
3 hours ago
$begingroup$
Consider $X,Y$ to be of trace 1 and their difference to be block diagonal $sigma^x,sigma^x$. Now $|X-Y|=$ identity matrix. Take the trace of both sides, the LHS gives 4 the RHS is 2.
$endgroup$
– lcv
3 hours ago
$begingroup$
@lcv. can you please elaborate your notations?
$endgroup$
– Samya Kumar Ray
2 hours ago
$begingroup$
@lcv. can you please elaborate your notations?
$endgroup$
– Samya Kumar Ray
2 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The answer is No. Here is a counter-example:
$$X=beginpmatrix 9 & 3 \ 3 & 1 endpmatrix,qquad Y=beginpmatrix 1 & 3 \ 3 & 9 endpmatrix.$$
answered 1 hour ago
Denis SerreDenis Serre
29.6k795197
$endgroup$
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: "504"
;
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%2fmathoverflow.net%2fquestions%2f326948%2fa-matrix-inequality-for-positive-definite-matrices%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
$begingroup$
The answer is No. Here is a counter-example:
$$X=beginpmatrix 9 & 3 \ 3 & 1 endpmatrix,qquad Y=beginpmatrix 1 & 3 \ 3 & 9 endpmatrix.$$
answered 1 hour ago
Denis SerreDenis Serre
29.6k795197
$endgroup$
add a comment |
$begingroup$
The answer is No. Here is a counter-example:
$$X=beginpmatrix 9 & 3 \ 3 & 1 endpmatrix,qquad Y=beginpmatrix 1 & 3 \ 3 & 9 endpmatrix.$$
answered 1 hour ago
Denis SerreDenis Serre
29.6k795197
$endgroup$
add a comment |
$begingroup$
The answer is No. Here is a counter-example:
$$X=beginpmatrix 9 & 3 \ 3 & 1 endpmatrix,qquad Y=beginpmatrix 1 & 3 \ 3 & 9 endpmatrix.$$
answered 1 hour ago
Denis SerreDenis Serre
29.6k795197
$endgroup$
The answer is No. Here is a counter-example:
$$X=beginpmatrix 9 & 3 \ 3 & 1 endpmatrix,qquad Y=beginpmatrix 1 & 3 \ 3 & 9 endpmatrix.$$
answered 1 hour ago
Denis SerreDenis Serre
29.6k795197
edited 17 mins ago
answered 1 hour ago
Denis SerreDenis Serre
29.6k795197
answered 1 hour ago
Denis SerreDenis Serre
29.6k795197
answered 1 hour ago
Denis SerreDenis Serre
29.6k795197
29.6k795197
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- 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%2fmathoverflow.net%2fquestions%2f326948%2fa-matrix-inequality-for-positive-definite-matrices%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
$begingroup$
Consider $X,Y$ to be of trace 1 and their difference to be block diagonal $sigma^x,sigma^x$. Now $|X-Y|=$ identity matrix. Take the trace of both sides, the LHS gives 4 the RHS is 2.
$endgroup$
– lcv
3 hours ago
$begingroup$
@lcv. can you please elaborate your notations?
$endgroup$
– Samya Kumar Ray
2 hours ago