lower bound for determinant of $(X^TY)$ Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)are there any bounds on the eigenvalues of products of positive-semidefinite matrices?Lower bound on Gram determinantLower bound on the smallest eigenvalueProve an upper bound for the determinant of a matrix ALower Bound for the induced matrix norm $||A||_infty,1$Simple lower bound for a determinantLower bound on absolute value of determinant of sum of matricesLower bounds for determinant of $AA^T$The Hadamard determinant problem: understanding a proof for the upper bound on matrix determinantsGreatest lower bound
Multi tool use
Illegal assignment from sObject to Id
What is this clumpy 20-30cm high yellow-flowered plant?
Dating a Former Employee
How were pictures turned from film to a big picture in a picture frame before digital scanning?
Is it a good idea to use CNN to classify 1D signal?
Is there any word for a place full of confusion?
Do wooden building fires get hotter than 600°C?
What is "gratricide"?
What's the meaning of "fortified infraction restraint"?
Sum letters are not two different
Why does it sometimes sound good to play a grace note as a lead in to a note in a melody?
Did Deadpool rescue all of the X-Force?
When a candle burns, why does the top of wick glow if bottom of flame is hottest?
As a beginner, should I get a Squier Strat with a SSS config or a HSS?
Why should I vote and accept answers?
What is the appropriate index architecture when forced to implement IsDeleted (soft deletes)?
Converted a Scalar function to a TVF function for parallel execution-Still running in Serial mode
Do any jurisdictions seriously consider reclassifying social media websites as publishers?
Performance gap between vector<bool> and array
Trademark violation for app?
How can I reduce the gap between left and right of cdot with a macro?
Maximum summed subsequences with non-adjacent items
How to write the following sign?
Why wasn't DOSKEY integrated with COMMAND.COM?
lower bound for determinant of $(X^TY)$
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)are there any bounds on the eigenvalues of products of positive-semidefinite matrices?Lower bound on Gram determinantLower bound on the smallest eigenvalueProve an upper bound for the determinant of a matrix ALower Bound for the induced matrix norm $||A||_infty,1$Simple lower bound for a determinantLower bound on absolute value of determinant of sum of matricesLower bounds for determinant of $AA^T$The Hadamard determinant problem: understanding a proof for the upper bound on matrix determinantsGreatest lower bound
$begingroup$
Am looking for a lower bound for determinant, $det(X^TY)$ where $X^T$ is $p times n$ and $Y$ is $n times p$. Is it $Tr(X^TY)^-1$? Regardless, what are other lower bounds for this? $X,Y$ are real-valued matrices.
linear-algebra matrices determinant inner-product-space upper-lower-bounds
asked Mar 27 at 16:42
hearsehearse
278
$endgroup$
add a comment |
$begingroup$
Am looking for a lower bound for determinant, $det(X^TY)$ where $X^T$ is $p times n$ and $Y$ is $n times p$. Is it $Tr(X^TY)^-1$? Regardless, what are other lower bounds for this? $X,Y$ are real-valued matrices.
linear-algebra matrices determinant inner-product-space upper-lower-bounds
asked Mar 27 at 16:42
hearsehearse
278
$endgroup$
$begingroup$
What is $Z$ in this context?
$endgroup$
– tch
Mar 27 at 17:30
$begingroup$
sry, typo. changed it to $Y$
$endgroup$
– hearse
Mar 27 at 17:31
$begingroup$
If $X = -cI$, $c>0$ and $Y=I$ when $c$ is negative and $n$ is odd then the $det(X^T)$ will be $-c^n$ while $1/Tr(X^TY)$ will be $-1/(cn)$. For $c$ large enough the determinant would be arbitrarily smaller than the bound from the trace.
$endgroup$
– tch
Mar 27 at 19:54
$begingroup$
You might want to check out en.wikipedia.org/wiki/Cauchy%E2%80%93Binet_formula
$endgroup$
– daw
Mar 27 at 20:36
$begingroup$
If $n < p$, then the determinant is always $0$, For $n ge p$, the lower bound of the absolute value of the determinant is $0$. The lower bound for the determinant itself is $-infty$. It is independent of the value of the trace for $p > 1$.
$endgroup$
– Paul Sinclair
Mar 28 at 0:38
add a comment |
$begingroup$
Am looking for a lower bound for determinant, $det(X^TY)$ where $X^T$ is $p times n$ and $Y$ is $n times p$. Is it $Tr(X^TY)^-1$? Regardless, what are other lower bounds for this? $X,Y$ are real-valued matrices.
linear-algebra matrices determinant inner-product-space upper-lower-bounds
asked Mar 27 at 16:42
hearsehearse
278
$endgroup$
Am looking for a lower bound for determinant, $det(X^TY)$ where $X^T$ is $p times n$ and $Y$ is $n times p$. Is it $Tr(X^TY)^-1$? Regardless, what are other lower bounds for this? $X,Y$ are real-valued matrices.
linear-algebra matrices determinant inner-product-space upper-lower-bounds
linear-algebra matrices determinant inner-product-space upper-lower-bounds
asked Mar 27 at 16:42
hearsehearse
278
asked Mar 27 at 16:42
hearsehearse
278
edited Mar 27 at 17:31
hearse
asked Mar 27 at 16:42
hearsehearse
278
asked Mar 27 at 16:42
hearsehearse
278
asked Mar 27 at 16:42
hearsehearse
278
278
$begingroup$
What is $Z$ in this context?
$endgroup$
– tch
Mar 27 at 17:30
$begingroup$
sry, typo. changed it to $Y$
$endgroup$
– hearse
Mar 27 at 17:31
$begingroup$
If $X = -cI$, $c>0$ and $Y=I$ when $c$ is negative and $n$ is odd then the $det(X^T)$ will be $-c^n$ while $1/Tr(X^TY)$ will be $-1/(cn)$. For $c$ large enough the determinant would be arbitrarily smaller than the bound from the trace.
$endgroup$
– tch
Mar 27 at 19:54
$begingroup$
You might want to check out en.wikipedia.org/wiki/Cauchy%E2%80%93Binet_formula
$endgroup$
– daw
Mar 27 at 20:36
$begingroup$
If $n < p$, then the determinant is always $0$, For $n ge p$, the lower bound of the absolute value of the determinant is $0$. The lower bound for the determinant itself is $-infty$. It is independent of the value of the trace for $p > 1$.
$endgroup$
– Paul Sinclair
Mar 28 at 0:38
add a comment |
$begingroup$
What is $Z$ in this context?
$endgroup$
– tch
Mar 27 at 17:30
$begingroup$
sry, typo. changed it to $Y$
$endgroup$
– hearse
Mar 27 at 17:31
$begingroup$
If $X = -cI$, $c>0$ and $Y=I$ when $c$ is negative and $n$ is odd then the $det(X^T)$ will be $-c^n$ while $1/Tr(X^TY)$ will be $-1/(cn)$. For $c$ large enough the determinant would be arbitrarily smaller than the bound from the trace.
$endgroup$
– tch
Mar 27 at 19:54
$begingroup$
You might want to check out en.wikipedia.org/wiki/Cauchy%E2%80%93Binet_formula
$endgroup$
– daw
Mar 27 at 20:36
$begingroup$
If $n < p$, then the determinant is always $0$, For $n ge p$, the lower bound of the absolute value of the determinant is $0$. The lower bound for the determinant itself is $-infty$. It is independent of the value of the trace for $p > 1$.
$endgroup$
– Paul Sinclair
Mar 28 at 0:38
$begingroup$
What is $Z$ in this context?
$endgroup$
– tch
Mar 27 at 17:30
$begingroup$
What is $Z$ in this context?
$endgroup$
– tch
Mar 27 at 17:30
$begingroup$
sry, typo. changed it to $Y$
$endgroup$
– hearse
Mar 27 at 17:31
$begingroup$
sry, typo. changed it to $Y$
$endgroup$
– hearse
Mar 27 at 17:31
$begingroup$
If $X = -cI$, $c>0$ and $Y=I$ when $c$ is negative and $n$ is odd then the $det(X^T)$ will be $-c^n$ while $1/Tr(X^TY)$ will be $-1/(cn)$. For $c$ large enough the determinant would be arbitrarily smaller than the bound from the trace.
$endgroup$
– tch
Mar 27 at 19:54
$begingroup$
If $X = -cI$, $c>0$ and $Y=I$ when $c$ is negative and $n$ is odd then the $det(X^T)$ will be $-c^n$ while $1/Tr(X^TY)$ will be $-1/(cn)$. For $c$ large enough the determinant would be arbitrarily smaller than the bound from the trace.
$endgroup$
– tch
Mar 27 at 19:54
$begingroup$
You might want to check out en.wikipedia.org/wiki/Cauchy%E2%80%93Binet_formula
$endgroup$
– daw
Mar 27 at 20:36
$begingroup$
You might want to check out en.wikipedia.org/wiki/Cauchy%E2%80%93Binet_formula
$endgroup$
– daw
Mar 27 at 20:36
$begingroup$
If $n < p$, then the determinant is always $0$, For $n ge p$, the lower bound of the absolute value of the determinant is $0$. The lower bound for the determinant itself is $-infty$. It is independent of the value of the trace for $p > 1$.
$endgroup$
– Paul Sinclair
Mar 28 at 0:38
$begingroup$
If $n < p$, then the determinant is always $0$, For $n ge p$, the lower bound of the absolute value of the determinant is $0$. The lower bound for the determinant itself is $-infty$. It is independent of the value of the trace for $p > 1$.
$endgroup$
– Paul Sinclair
Mar 28 at 0:38
add a comment |
0
active
oldest
votes
Your Answer
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%2f3164745%2flower-bound-for-determinant-of-xty%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.
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%2f3164745%2flower-bound-for-determinant-of-xty%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
7bLjv Sji3fwG0BaLyOiuR9aFOsrZV
$begingroup$
What is $Z$ in this context?
$endgroup$
– tch
Mar 27 at 17:30
$begingroup$
sry, typo. changed it to $Y$
$endgroup$
– hearse
Mar 27 at 17:31
$begingroup$
If $X = -cI$, $c>0$ and $Y=I$ when $c$ is negative and $n$ is odd then the $det(X^T)$ will be $-c^n$ while $1/Tr(X^TY)$ will be $-1/(cn)$. For $c$ large enough the determinant would be arbitrarily smaller than the bound from the trace.
$endgroup$
– tch
Mar 27 at 19:54
$begingroup$
You might want to check out en.wikipedia.org/wiki/Cauchy%E2%80%93Binet_formula
$endgroup$
– daw
Mar 27 at 20:36
$begingroup$
If $n < p$, then the determinant is always $0$, For $n ge p$, the lower bound of the absolute value of the determinant is $0$. The lower bound for the determinant itself is $-infty$. It is independent of the value of the trace for $p > 1$.
$endgroup$
– Paul Sinclair
Mar 28 at 0:38