Simplification of $x = left(A + sum_i alpha_i B_iright)^-1 A y$, $A succ 0, A in M_n,n(mathbbC)$, $textrmrank(B_i) = 1$

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition.



Problem definition:
Say $A succ 0, A in M_n,nleft( mathbbC right)$, is Hermitian and full-rank (positive definite) matrix. $B_i in M_n,nleft( mathbbC right)$ and rank-$1$, $alpha_i in mathbbC$ some constant, $y in M_n,1left( mathbbC right)$ some column vector.



beginalign
x = left(A + sum_i=0^m alpha_i B_iright)^-1 A y
endalign



Question: Can this matrix inversion be simplified such that big matrix inversion can be avoided (since $n approx 8$k)? Thank you in advance.



I can't make any progress with matrix inversion lemmas, e.g., in matrix cookbook.




linear-algebra matrices inverse




share|cite|improve this question
















  • 1




    does this (math.stackexchange.com/questions/17776/…) help you at all?
    – bobcliffe
    Aug 23 at 9:01











  • Thanks, I will try that.
    – user550103
    Aug 23 at 9:10














up vote
0
down vote

favorite












I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition.



Problem definition:
Say $A succ 0, A in M_n,nleft( mathbbC right)$, is Hermitian and full-rank (positive definite) matrix. $B_i in M_n,nleft( mathbbC right)$ and rank-$1$, $alpha_i in mathbbC$ some constant, $y in M_n,1left( mathbbC right)$ some column vector.



beginalign
x = left(A + sum_i=0^m alpha_i B_iright)^-1 A y
endalign



Question: Can this matrix inversion be simplified such that big matrix inversion can be avoided (since $n approx 8$k)? Thank you in advance.



I can't make any progress with matrix inversion lemmas, e.g., in matrix cookbook.




linear-algebra matrices inverse




share|cite|improve this question
















  • 1




    does this (math.stackexchange.com/questions/17776/…) help you at all?
    – bobcliffe
    Aug 23 at 9:01











  • Thanks, I will try that.
    – user550103
    Aug 23 at 9:10












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition.



Problem definition:
Say $A succ 0, A in M_n,nleft( mathbbC right)$, is Hermitian and full-rank (positive definite) matrix. $B_i in M_n,nleft( mathbbC right)$ and rank-$1$, $alpha_i in mathbbC$ some constant, $y in M_n,1left( mathbbC right)$ some column vector.



beginalign
x = left(A + sum_i=0^m alpha_i B_iright)^-1 A y
endalign



Question: Can this matrix inversion be simplified such that big matrix inversion can be avoided (since $n approx 8$k)? Thank you in advance.



I can't make any progress with matrix inversion lemmas, e.g., in matrix cookbook.




linear-algebra matrices inverse




share|cite|improve this question












I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition.



Problem definition:
Say $A succ 0, A in M_n,nleft( mathbbC right)$, is Hermitian and full-rank (positive definite) matrix. $B_i in M_n,nleft( mathbbC right)$ and rank-$1$, $alpha_i in mathbbC$ some constant, $y in M_n,1left( mathbbC right)$ some column vector.



beginalign
x = left(A + sum_i=0^m alpha_i B_iright)^-1 A y
endalign



Question: Can this matrix inversion be simplified such that big matrix inversion can be avoided (since $n approx 8$k)? Thank you in advance.



I can't make any progress with matrix inversion lemmas, e.g., in matrix cookbook.





linear-algebra matrices inverse





share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 23 at 8:29









user550103

549213




549213







  • 1




    does this (math.stackexchange.com/questions/17776/…) help you at all?
    – bobcliffe
    Aug 23 at 9:01











  • Thanks, I will try that.
    – user550103
    Aug 23 at 9:10












  • 1




    does this (math.stackexchange.com/questions/17776/…) help you at all?
    – bobcliffe
    Aug 23 at 9:01











  • Thanks, I will try that.
    – user550103
    Aug 23 at 9:10







1




1




does this (math.stackexchange.com/questions/17776/…) help you at all?
– bobcliffe
Aug 23 at 9:01





does this (math.stackexchange.com/questions/17776/…) help you at all?
– bobcliffe
Aug 23 at 9:01













Thanks, I will try that.
– user550103
Aug 23 at 9:10




Thanks, I will try that.
– user550103
Aug 23 at 9:10















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',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);













 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2891864%2fsimplification-of-x-lefta-sum-i-alpha-i-b-i-right-1-a-y-a-suc%23new-answer', 'question_page');

);

Post as a guest






















active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes















 

draft saved


draft discarded















































 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2891864%2fsimplification-of-x-lefta-sum-i-alpha-i-b-i-right-1-a-y-a-suc%23new-answer', 'question_page');

);

Post as a guest
































































-

這個網誌中的熱門文章

How to combine Bézier curves to a surface?

圖片
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier cur...
閱讀完整內容

Mutual Information Always Non-negative

圖片
Clash Royale CLAN TAG #URR8PPP up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 '12 at 15:38 Cameron Buie 83.6k 7 71 154 asked Jun 17 '12 at 15:30 Omri 359 3 13 3 Convexity of the function $tmapsto tlog t$. – Did Jun 17 '12 at 15:33 In addition, the convexity properties require the coefficients in the linear combination sum 1. Then, as p(x,y) is a probability distribution, it fullfits such condition. – Francisco Javier Delgado Ceped Aug 10 at 18:44 add a comment  |  up vote 10 down vote favorite 6 What is the simplest proof that mutual information is always non-negative? i.e., $I(X;Y)ge0$ probability-theory share | cite | improve this question edited Jun 17 ...
閱讀完整內容

Why am i infinitely getting the same tweet with the Twitter Search API?

圖片
up vote 0 down vote favorite My code workes perfectly for other queries, but for one query I am infinitely getting the same tweet. I can't understand what the problem is. API call: query='Allergic asthma OR Nonallergic asthma OR Occupational asthma OR EIB OR Exercise-induced bronchoconstriction OR Nocturnal asthma OR Cough-variant asthma' new_tweets = api.search(q=searchQuery, count=100, since_id=since_id, lang='en',tweet_mode='extended') if not new_tweets: print("No more tweets found") break for tweet in new_tweets: if True: print(jsonpickle.encode(tweet._json, unpicklable=False)) my_file = open('searchTweets.txt','a') my_file.write(jsonpickle.encode(tweet._json, unpicklable=False)+'n') my_file.close() else: continue Output in file: https://pastebin.com/JmRCsTeE These are the JSONs of the same Tweet. Other queries that work: Breast cancer OR Prostate cancer OR Colon cancer OR Lung cancer Type 2 dia...
閱讀完整內容