How to prove Wielandt minimax formula?
The statements are as follows:
Let $1leqslant i_1<i_2<cdots<i_kleqslant n$ be integers. Define a partial flag to be a nested collection $V_1subset V_2cdotssubset V_k$ of subspaces of $mathbb{C}^n$ s.t. $dim(V_j)=i_j$ for all j between 1 and k. Define the Schubert variety $X(V_1,...,V_k)$ to be the collection of all k-dimensional subspaces W s.t. $dim(Wcap V_j)geqslant j$. Show that for any Hermitian matrix A,
$$lambda_{i_1}+...+lambda_{i_k}=sup_{V_1,...,V_k} inf_{Win X(V_1,...,V_k)}tr(A|_{W})$$.
Where $lambda_{i_j}$ means the $i_j$ th eigenvalue of A, from large to small. The trace in the right formula stands for the partial trace of A on W.
I have proofed that the LHS is no bigger than the RHS using induction on k, but I do not know how to prove the other side.
linear-algebra eigenvalues-eigenvectors random-matrices schubert-calculus
edited 2 days ago
Matt Samuel
37.5k63565
asked Jul 26 '18 at 16:25
user579758user579758
11
add a comment |
The statements are as follows:
Let $1leqslant i_1<i_2<cdots<i_kleqslant n$ be integers. Define a partial flag to be a nested collection $V_1subset V_2cdotssubset V_k$ of subspaces of $mathbb{C}^n$ s.t. $dim(V_j)=i_j$ for all j between 1 and k. Define the Schubert variety $X(V_1,...,V_k)$ to be the collection of all k-dimensional subspaces W s.t. $dim(Wcap V_j)geqslant j$. Show that for any Hermitian matrix A,
$$lambda_{i_1}+...+lambda_{i_k}=sup_{V_1,...,V_k} inf_{Win X(V_1,...,V_k)}tr(A|_{W})$$.
Where $lambda_{i_j}$ means the $i_j$ th eigenvalue of A, from large to small. The trace in the right formula stands for the partial trace of A on W.
I have proofed that the LHS is no bigger than the RHS using induction on k, but I do not know how to prove the other side.
linear-algebra eigenvalues-eigenvectors random-matrices schubert-calculus
edited 2 days ago
Matt Samuel
37.5k63565
asked Jul 26 '18 at 16:25
user579758user579758
11
add a comment |
The statements are as follows:
Let $1leqslant i_1<i_2<cdots<i_kleqslant n$ be integers. Define a partial flag to be a nested collection $V_1subset V_2cdotssubset V_k$ of subspaces of $mathbb{C}^n$ s.t. $dim(V_j)=i_j$ for all j between 1 and k. Define the Schubert variety $X(V_1,...,V_k)$ to be the collection of all k-dimensional subspaces W s.t. $dim(Wcap V_j)geqslant j$. Show that for any Hermitian matrix A,
$$lambda_{i_1}+...+lambda_{i_k}=sup_{V_1,...,V_k} inf_{Win X(V_1,...,V_k)}tr(A|_{W})$$.
Where $lambda_{i_j}$ means the $i_j$ th eigenvalue of A, from large to small. The trace in the right formula stands for the partial trace of A on W.
I have proofed that the LHS is no bigger than the RHS using induction on k, but I do not know how to prove the other side.
linear-algebra eigenvalues-eigenvectors random-matrices schubert-calculus
edited 2 days ago
Matt Samuel
37.5k63565
asked Jul 26 '18 at 16:25
user579758user579758
11
The statements are as follows:
Let $1leqslant i_1<i_2<cdots<i_kleqslant n$ be integers. Define a partial flag to be a nested collection $V_1subset V_2cdotssubset V_k$ of subspaces of $mathbb{C}^n$ s.t. $dim(V_j)=i_j$ for all j between 1 and k. Define the Schubert variety $X(V_1,...,V_k)$ to be the collection of all k-dimensional subspaces W s.t. $dim(Wcap V_j)geqslant j$. Show that for any Hermitian matrix A,
$$lambda_{i_1}+...+lambda_{i_k}=sup_{V_1,...,V_k} inf_{Win X(V_1,...,V_k)}tr(A|_{W})$$.
Where $lambda_{i_j}$ means the $i_j$ th eigenvalue of A, from large to small. The trace in the right formula stands for the partial trace of A on W.
I have proofed that the LHS is no bigger than the RHS using induction on k, but I do not know how to prove the other side.
linear-algebra eigenvalues-eigenvectors random-matrices schubert-calculus
linear-algebra eigenvalues-eigenvectors random-matrices schubert-calculus
edited 2 days ago
Matt Samuel
37.5k63565
asked Jul 26 '18 at 16:25
user579758user579758
11
edited 2 days ago
Matt Samuel
37.5k63565
asked Jul 26 '18 at 16:25
user579758user579758
11
edited 2 days ago
Matt Samuel
37.5k63565
edited 2 days ago
Matt Samuel
37.5k63565
edited 2 days ago
Matt Samuel
37.5k63565
37.5k63565
asked Jul 26 '18 at 16:25
user579758user579758
11
asked Jul 26 '18 at 16:25
user579758user579758
11
asked Jul 26 '18 at 16:25
user579758user579758
11
11
add a comment |
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%2f2863568%2fhow-to-prove-wielandt-minimax-formula%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%2f2863568%2fhow-to-prove-wielandt-minimax-formula%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
