Is uniqueness of steady-state vector sufficient to regular transition matrix?












0















If P is a transition matrix, then a steady-state vector for is a probability vector q such that $Pmathrm{q}=mathrm{q}$.



A transition matrix P is regular if some power $P^k$ contain only strictly positive entries.




We know that the steady-state vector is the eigenvector of P associated with the eignvalue 1.



So if rank(P-I)=Dim(P)-1, then the eigenvector of P which column sum is 1 (steady-state vector) will be unique.



Is uniqueness of steady-state vector sufficient to the regularness of P?










share|cite|improve this question



























    0















    If P is a transition matrix, then a steady-state vector for is a probability vector q such that $Pmathrm{q}=mathrm{q}$.



    A transition matrix P is regular if some power $P^k$ contain only strictly positive entries.




    We know that the steady-state vector is the eigenvector of P associated with the eignvalue 1.



    So if rank(P-I)=Dim(P)-1, then the eigenvector of P which column sum is 1 (steady-state vector) will be unique.



    Is uniqueness of steady-state vector sufficient to the regularness of P?










    share|cite|improve this question

























      0












      0








      0








      If P is a transition matrix, then a steady-state vector for is a probability vector q such that $Pmathrm{q}=mathrm{q}$.



      A transition matrix P is regular if some power $P^k$ contain only strictly positive entries.




      We know that the steady-state vector is the eigenvector of P associated with the eignvalue 1.



      So if rank(P-I)=Dim(P)-1, then the eigenvector of P which column sum is 1 (steady-state vector) will be unique.



      Is uniqueness of steady-state vector sufficient to the regularness of P?










      share|cite|improve this question














      If P is a transition matrix, then a steady-state vector for is a probability vector q such that $Pmathrm{q}=mathrm{q}$.



      A transition matrix P is regular if some power $P^k$ contain only strictly positive entries.




      We know that the steady-state vector is the eigenvector of P associated with the eignvalue 1.



      So if rank(P-I)=Dim(P)-1, then the eigenvector of P which column sum is 1 (steady-state vector) will be unique.



      Is uniqueness of steady-state vector sufficient to the regularness of P?







      linear-algebra markov-chains






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 30 '18 at 18:38









      Charles BaoCharles Bao

      751919




      751919






















          1 Answer
          1






          active

          oldest

          votes


















          0














          Have already found a counterexample.



          $$P_2=begin{pmatrix}
          0 & 0.5 & 0.5 & 0\
          0.5 & 0 & 0 & 0.5\
          0.5 & 0 & 0 & 0.5\
          0 & 0.5 & 0.5 & 0\
          end{pmatrix}$$



          with unique steady-state vector $(0.25,0.25,0.25,0.25)^T$






          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',
            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
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3020471%2fis-uniqueness-of-steady-state-vector-sufficient-to-regular-transition-matrix%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            0














            Have already found a counterexample.



            $$P_2=begin{pmatrix}
            0 & 0.5 & 0.5 & 0\
            0.5 & 0 & 0 & 0.5\
            0.5 & 0 & 0 & 0.5\
            0 & 0.5 & 0.5 & 0\
            end{pmatrix}$$



            with unique steady-state vector $(0.25,0.25,0.25,0.25)^T$






            share|cite|improve this answer


























              0














              Have already found a counterexample.



              $$P_2=begin{pmatrix}
              0 & 0.5 & 0.5 & 0\
              0.5 & 0 & 0 & 0.5\
              0.5 & 0 & 0 & 0.5\
              0 & 0.5 & 0.5 & 0\
              end{pmatrix}$$



              with unique steady-state vector $(0.25,0.25,0.25,0.25)^T$






              share|cite|improve this answer
























                0












                0








                0






                Have already found a counterexample.



                $$P_2=begin{pmatrix}
                0 & 0.5 & 0.5 & 0\
                0.5 & 0 & 0 & 0.5\
                0.5 & 0 & 0 & 0.5\
                0 & 0.5 & 0.5 & 0\
                end{pmatrix}$$



                with unique steady-state vector $(0.25,0.25,0.25,0.25)^T$






                share|cite|improve this answer












                Have already found a counterexample.



                $$P_2=begin{pmatrix}
                0 & 0.5 & 0.5 & 0\
                0.5 & 0 & 0 & 0.5\
                0.5 & 0 & 0 & 0.5\
                0 & 0.5 & 0.5 & 0\
                end{pmatrix}$$



                with unique steady-state vector $(0.25,0.25,0.25,0.25)^T$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 30 '18 at 19:10









                Charles BaoCharles Bao

                751919




                751919






























                    draft saved

                    draft discarded




















































                    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.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3020471%2fis-uniqueness-of-steady-state-vector-sufficient-to-regular-transition-matrix%23new-answer', 'question_page');
                    }
                    );

                    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







                    Popular posts from this blog

                    Probability when a professor distributes a quiz and homework assignment to a class of n students.

                    Aardman Animations

                    Are they similar matrix