Trace of a product of two positive definite matrices
$begingroup$
Let $A, B, C_t in mathbb{R}^{n times n}$ be positive definite matrixand $C_t$ is defined as
begin{align*}
C_{t,(i,j)}=begin{cases}
A_{i,j}, &i,j < t \
B_{i,j}, &i,j ge t \
0, &text{Otherwise}
end{cases}
end{align*}
I guess the $Tr(AC_t^{-1})$ is monotonically increasing or decreasing over $t$.
But I have difficulty to prove it, any hints?
linear-algebra trace positive-definite block-matrices
asked Dec 18 '18 at 10:04
kw1924kw1924
112
$endgroup$
add a comment |
$begingroup$
Let $A, B, C_t in mathbb{R}^{n times n}$ be positive definite matrixand $C_t$ is defined as
begin{align*}
C_{t,(i,j)}=begin{cases}
A_{i,j}, &i,j < t \
B_{i,j}, &i,j ge t \
0, &text{Otherwise}
end{cases}
end{align*}
I guess the $Tr(AC_t^{-1})$ is monotonically increasing or decreasing over $t$.
But I have difficulty to prove it, any hints?
linear-algebra trace positive-definite block-matrices
asked Dec 18 '18 at 10:04
kw1924kw1924
112
$endgroup$
$begingroup$
It obviously fails already for diagonal matrices $A,B$.
$endgroup$
– metamorphy
Dec 18 '18 at 10:39
add a comment |
$begingroup$
Let $A, B, C_t in mathbb{R}^{n times n}$ be positive definite matrixand $C_t$ is defined as
begin{align*}
C_{t,(i,j)}=begin{cases}
A_{i,j}, &i,j < t \
B_{i,j}, &i,j ge t \
0, &text{Otherwise}
end{cases}
end{align*}
I guess the $Tr(AC_t^{-1})$ is monotonically increasing or decreasing over $t$.
But I have difficulty to prove it, any hints?
linear-algebra trace positive-definite block-matrices
asked Dec 18 '18 at 10:04
kw1924kw1924
112
$endgroup$
Let $A, B, C_t in mathbb{R}^{n times n}$ be positive definite matrixand $C_t$ is defined as
begin{align*}
C_{t,(i,j)}=begin{cases}
A_{i,j}, &i,j < t \
B_{i,j}, &i,j ge t \
0, &text{Otherwise}
end{cases}
end{align*}
I guess the $Tr(AC_t^{-1})$ is monotonically increasing or decreasing over $t$.
But I have difficulty to prove it, any hints?
linear-algebra trace positive-definite block-matrices
linear-algebra trace positive-definite block-matrices
asked Dec 18 '18 at 10:04
kw1924kw1924
112
asked Dec 18 '18 at 10:04
kw1924kw1924
112
asked Dec 18 '18 at 10:04
kw1924kw1924
112
asked Dec 18 '18 at 10:04
kw1924kw1924
112
asked Dec 18 '18 at 10:04
kw1924kw1924
112
112
$begingroup$
It obviously fails already for diagonal matrices $A,B$.
$endgroup$
– metamorphy
Dec 18 '18 at 10:39
add a comment |
$begingroup$
It obviously fails already for diagonal matrices $A,B$.
$endgroup$
– metamorphy
Dec 18 '18 at 10:39
$begingroup$
It obviously fails already for diagonal matrices $A,B$.
$endgroup$
– metamorphy
Dec 18 '18 at 10:39
$begingroup$
It obviously fails already for diagonal matrices $A,B$.
$endgroup$
– metamorphy
Dec 18 '18 at 10:39
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%2f3044978%2ftrace-of-a-product-of-two-positive-definite-matrices%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%2f3044978%2ftrace-of-a-product-of-two-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
$begingroup$
It obviously fails already for diagonal matrices $A,B$.
$endgroup$
– metamorphy
Dec 18 '18 at 10:39