Orthogonality of stochastic matrix

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











up vote
0
down vote

favorite












Given a column stochastic matrix $P$, I wanted to give a relation between $|P|$ and orthogonality of $P$.



One simple way to think about how close $P$ is to being orthogonal is $|P^topP - I|$. Then, I simply went ahead and wrote:
$$|P| leq sqrt$$



First of all, does the above make sense? Secondly, I am not utilizing the fact that $P$ is stochastic, are there ways to get any better relationship than the one given above? Are there any measures defined for stochastic matrices (in literature) that measure how close a matrix is to being orthogonal?



P.S. All norms are Frobenius norms.




matrices orthogonal-matrices stochastic-matrices




share|cite|improve this question


















  • 1




    For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
    – Algebraic Pavel
    Oct 26 '15 at 16:28











  • @AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
    – TenaliRaman
    Oct 26 '15 at 17:27














up vote
0
down vote

favorite












Given a column stochastic matrix $P$, I wanted to give a relation between $|P|$ and orthogonality of $P$.



One simple way to think about how close $P$ is to being orthogonal is $|P^topP - I|$. Then, I simply went ahead and wrote:
$$|P| leq sqrt$$



First of all, does the above make sense? Secondly, I am not utilizing the fact that $P$ is stochastic, are there ways to get any better relationship than the one given above? Are there any measures defined for stochastic matrices (in literature) that measure how close a matrix is to being orthogonal?



P.S. All norms are Frobenius norms.




matrices orthogonal-matrices stochastic-matrices




share|cite|improve this question


















  • 1




    For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
    – Algebraic Pavel
    Oct 26 '15 at 16:28











  • @AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
    – TenaliRaman
    Oct 26 '15 at 17:27












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Given a column stochastic matrix $P$, I wanted to give a relation between $|P|$ and orthogonality of $P$.



One simple way to think about how close $P$ is to being orthogonal is $|P^topP - I|$. Then, I simply went ahead and wrote:
$$|P| leq sqrt$$



First of all, does the above make sense? Secondly, I am not utilizing the fact that $P$ is stochastic, are there ways to get any better relationship than the one given above? Are there any measures defined for stochastic matrices (in literature) that measure how close a matrix is to being orthogonal?



P.S. All norms are Frobenius norms.




matrices orthogonal-matrices stochastic-matrices




share|cite|improve this question














Given a column stochastic matrix $P$, I wanted to give a relation between $|P|$ and orthogonality of $P$.



One simple way to think about how close $P$ is to being orthogonal is $|P^topP - I|$. Then, I simply went ahead and wrote:
$$|P| leq sqrt$$



First of all, does the above make sense? Secondly, I am not utilizing the fact that $P$ is stochastic, are there ways to get any better relationship than the one given above? Are there any measures defined for stochastic matrices (in literature) that measure how close a matrix is to being orthogonal?



P.S. All norms are Frobenius norms.





matrices orthogonal-matrices stochastic-matrices





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 17 at 7:14









Rodrigo de Azevedo

12.6k41751




12.6k41751










asked Oct 26 '15 at 14:26









TenaliRaman

3,0891222




3,0891222







  • 1




    For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
    – Algebraic Pavel
    Oct 26 '15 at 16:28











  • @AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
    – TenaliRaman
    Oct 26 '15 at 17:27












  • 1




    For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
    – Algebraic Pavel
    Oct 26 '15 at 16:28











  • @AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
    – TenaliRaman
    Oct 26 '15 at 17:27







1




1




For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
– Algebraic Pavel
Oct 26 '15 at 16:28





For the loss of orthogonality, you might find some interesting pointers in Section 4 here.
– Algebraic Pavel
Oct 26 '15 at 16:28













@AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
– TenaliRaman
Oct 26 '15 at 17:27




@AlgebraicPavel Thanks for the pointer! Theorem 4.1 looks very very useful!!
– TenaliRaman
Oct 26 '15 at 17:27















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%2f1498507%2forthogonality-of-stochastic-matrix%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%2f1498507%2forthogonality-of-stochastic-matrix%23new-answer', 'question_page');

);

Post as a guest
































































-

這個網誌中的熱門文章

Is there any way to eliminate the singular point to solve this integral by hand or by approximations?

圖片
Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite 1 $$int_0^1 (1+(x^-p-1)^1-p)^frac1pdx, pin mathbbR, pge 1$$ I find that $x=0,1$ are singular points(discontinuous points of this integral), is there a way to eliminate those singular points to make sure that this integral can be calculated by hand or by Taylor approximations. In either cases, what are those steps of calculating the integral? calculus real-analysis analysis approximation share | cite | improve this question edited Sep 11 at 5:07 asked Sep 11 at 4:48 Mclalalala 126 8 1 What is $p$? Please give more information to make it complete. – xbh Sep 11 at 5:02 @xbh already fixed it, thanks for reminding. – Mclalalala Sep 11 at 5:10 Seems that $0$ is not singular here… [I might compute wrong] – xbh Sep 11 at 5:30 @xbh can you show me how to get the close formula for the integral above? – Mclalalala Sep 11 at 5:33
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

Amount of Number Combinations to Reach a Sum of 10 With Integers 1-9 Using 2 or More Integers [closed]

圖片
Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite I received a word problem that goes like this. A local kindergarten is thinking of making posters that show all the different ways of adding two or more integers from 1 to 9 to get a sum of 10. If there is enough space on each poster for up to 50 possible solutions, how many posters will the school need to make? (Note: sums that contain the same number but in a different order are considered to be different; for example, 1 + 9 and 9 + 1 are two different solutions.) What is the answer to this problem, and more importantly, how do I solve it? combinatorics combinations share | cite | improve this question edited Sep 10 at 23:54 N. F. Taussig 39.9k 9 32 53 asked Sep 10 at 21:43 Jolly 3 1 closed as off-topic by Theoretical Economist, Adrian Keister, user99914, Xander Henderson, Deepesh Meena Sep 11 at 3:26 This question appears to be off-topic. The users who voted to clos
閱讀完整內容