positive-definite matrix with equal row sums

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











up vote
0
down vote

favorite












Consider the following square matrix
beginbmatrix
a_1,1 & a_1,2 & a_1,3 & a_1,4 \
a_2,1 & a_2,2 & a_2,3 & a_2,4 \
a_3,1 & a_3,2 & a_3,3 & a_3,4 \
a_4,1 & a_4,2 & a_4,3 & a_4,4
endbmatrix
where $0leq a_i,j leq 1$ for all $i,j in [1,16]$. Consider some $delta >0$, I know that this matrix satisfies
beginalign
left| sum_j=1^4 a_i,j - frac14 sum_i=1^4 sum_j=1^4 a_i,j right| leq delta quad textfor all i in [1,4],
endalign
that is, all the row sums almost equal to the sum of all elements divided by the number of rows (or columns).



QUESTION: The previous property implies
beginalign
left| sum_i=1^4 a_i,j - frac14 sum_i=1^4 sum_j=1^4 a_i,j right| leq delta quad textfor all j in [1,4],
endalign
?????????




matrices symmetric-matrices




share|cite|improve this question

























    up vote
    0
    down vote

    favorite












    Consider the following square matrix
    beginbmatrix
    a_1,1 & a_1,2 & a_1,3 & a_1,4 \
    a_2,1 & a_2,2 & a_2,3 & a_2,4 \
    a_3,1 & a_3,2 & a_3,3 & a_3,4 \
    a_4,1 & a_4,2 & a_4,3 & a_4,4
    endbmatrix
    where $0leq a_i,j leq 1$ for all $i,j in [1,16]$. Consider some $delta >0$, I know that this matrix satisfies
    beginalign
    left| sum_j=1^4 a_i,j - frac14 sum_i=1^4 sum_j=1^4 a_i,j right| leq delta quad textfor all i in [1,4],
    endalign
    that is, all the row sums almost equal to the sum of all elements divided by the number of rows (or columns).



    QUESTION: The previous property implies
    beginalign
    left| sum_i=1^4 a_i,j - frac14 sum_i=1^4 sum_j=1^4 a_i,j right| leq delta quad textfor all j in [1,4],
    endalign
    ?????????




    matrices symmetric-matrices




    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Consider the following square matrix
      beginbmatrix
      a_1,1 & a_1,2 & a_1,3 & a_1,4 \
      a_2,1 & a_2,2 & a_2,3 & a_2,4 \
      a_3,1 & a_3,2 & a_3,3 & a_3,4 \
      a_4,1 & a_4,2 & a_4,3 & a_4,4
      endbmatrix
      where $0leq a_i,j leq 1$ for all $i,j in [1,16]$. Consider some $delta >0$, I know that this matrix satisfies
      beginalign
      left| sum_j=1^4 a_i,j - frac14 sum_i=1^4 sum_j=1^4 a_i,j right| leq delta quad textfor all i in [1,4],
      endalign
      that is, all the row sums almost equal to the sum of all elements divided by the number of rows (or columns).



      QUESTION: The previous property implies
      beginalign
      left| sum_i=1^4 a_i,j - frac14 sum_i=1^4 sum_j=1^4 a_i,j right| leq delta quad textfor all j in [1,4],
      endalign
      ?????????




      matrices symmetric-matrices




      share|cite|improve this question













      Consider the following square matrix
      beginbmatrix
      a_1,1 & a_1,2 & a_1,3 & a_1,4 \
      a_2,1 & a_2,2 & a_2,3 & a_2,4 \
      a_3,1 & a_3,2 & a_3,3 & a_3,4 \
      a_4,1 & a_4,2 & a_4,3 & a_4,4
      endbmatrix
      where $0leq a_i,j leq 1$ for all $i,j in [1,16]$. Consider some $delta >0$, I know that this matrix satisfies
      beginalign
      left| sum_j=1^4 a_i,j - frac14 sum_i=1^4 sum_j=1^4 a_i,j right| leq delta quad textfor all i in [1,4],
      endalign
      that is, all the row sums almost equal to the sum of all elements divided by the number of rows (or columns).



      QUESTION: The previous property implies
      beginalign
      left| sum_i=1^4 a_i,j - frac14 sum_i=1^4 sum_j=1^4 a_i,j right| leq delta quad textfor all j in [1,4],
      endalign
      ?????????





      matrices symmetric-matrices





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 17 at 10:19
























      asked Jul 17 at 10:12









      Jaume

      315




      315




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote













          No, there are easy counter examples:
          $$
          beginbmatrix
          1&1&0&0\
          0&0&0&0\
          0&0&1&0\
          0&0&0&1
          endbmatrix$$






          share|cite|improve this answer





















            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%2f2854355%2fpositive-definite-matrix-with-equal-row-sums%23new-answer', 'question_page');

            );

            Post as a guest

















            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote













            No, there are easy counter examples:
            $$
            beginbmatrix
            1&1&0&0\
            0&0&0&0\
            0&0&1&0\
            0&0&0&1
            endbmatrix$$






            share|cite|improve this answer

























              up vote
              1
              down vote













              No, there are easy counter examples:
              $$
              beginbmatrix
              1&1&0&0\
              0&0&0&0\
              0&0&1&0\
              0&0&0&1
              endbmatrix$$






              share|cite|improve this answer























                up vote
                1
                down vote










                up vote
                1
                down vote









                No, there are easy counter examples:
                $$
                beginbmatrix
                1&1&0&0\
                0&0&0&0\
                0&0&1&0\
                0&0&0&1
                endbmatrix$$






                share|cite|improve this answer













                No, there are easy counter examples:
                $$
                beginbmatrix
                1&1&0&0\
                0&0&0&0\
                0&0&1&0\
                0&0&0&1
                endbmatrix$$







                share|cite|improve this answer













                share|cite|improve this answer



                share|cite|improve this answer











                answered Jul 17 at 10:17









                Jürg Merlin Spaak

                81613




                81613






















                     

                    draft saved


                    draft discarded


























                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2854355%2fpositive-definite-matrix-with-equal-row-sums%23new-answer', 'question_page');

                    );

                    Post as a guest
































































                    Comments

                    Post a Comment

                    Popular posts from this blog

                    What is the equation of a 3D cone with generalised tilt?

                    Image
                    Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
                    Read more

                    Color the edges and diagonals of a regular polygon

                    Image
                    Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
                    Read more

                    Relationship between determinant of matrix and determinant of adjoint?

                    Image
                    Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
                    Read more