Recurrence relations in Matrices raised to natural powers












0












$begingroup$


Let some matrix be $A$, what methods are there for attaining a recurrence relation which allows us to calculate - for example - the leading term in the matrix $A^n$ when given the leading terms of a sufficient amount of previous matrices $A^{n-1}, A^{n-2},...$



I ask this because I was studying the leading term in the matrix $A^n$ where $$A=
begin{bmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
7 & 8 & 9 \
end{bmatrix}
$$

which can be found in a ridiculous closed form (https://oeis.org/A321045), but has the following recurrence relation between the nth powers of the matrix:
$$a(n+2)=15a(n+1)+18a(n)$$
for $n ge 1$. How does one find a relation like this or equivalently find it's corresponding generating function - $frac{3x^2+14x-1}{18x^2+15x-1}$ ?










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    this is just Cayley-Hamilton. In your case, the determinant is zero, so you don't get the full effect, with would be expected to relate $n+3,n+2,n+1,n$
    $endgroup$
    – Will Jagy
    Jan 8 at 23:25
















0












$begingroup$


Let some matrix be $A$, what methods are there for attaining a recurrence relation which allows us to calculate - for example - the leading term in the matrix $A^n$ when given the leading terms of a sufficient amount of previous matrices $A^{n-1}, A^{n-2},...$



I ask this because I was studying the leading term in the matrix $A^n$ where $$A=
begin{bmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
7 & 8 & 9 \
end{bmatrix}
$$

which can be found in a ridiculous closed form (https://oeis.org/A321045), but has the following recurrence relation between the nth powers of the matrix:
$$a(n+2)=15a(n+1)+18a(n)$$
for $n ge 1$. How does one find a relation like this or equivalently find it's corresponding generating function - $frac{3x^2+14x-1}{18x^2+15x-1}$ ?










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    this is just Cayley-Hamilton. In your case, the determinant is zero, so you don't get the full effect, with would be expected to relate $n+3,n+2,n+1,n$
    $endgroup$
    – Will Jagy
    Jan 8 at 23:25














0












0








0





$begingroup$


Let some matrix be $A$, what methods are there for attaining a recurrence relation which allows us to calculate - for example - the leading term in the matrix $A^n$ when given the leading terms of a sufficient amount of previous matrices $A^{n-1}, A^{n-2},...$



I ask this because I was studying the leading term in the matrix $A^n$ where $$A=
begin{bmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
7 & 8 & 9 \
end{bmatrix}
$$

which can be found in a ridiculous closed form (https://oeis.org/A321045), but has the following recurrence relation between the nth powers of the matrix:
$$a(n+2)=15a(n+1)+18a(n)$$
for $n ge 1$. How does one find a relation like this or equivalently find it's corresponding generating function - $frac{3x^2+14x-1}{18x^2+15x-1}$ ?










share|cite|improve this question









$endgroup$




Let some matrix be $A$, what methods are there for attaining a recurrence relation which allows us to calculate - for example - the leading term in the matrix $A^n$ when given the leading terms of a sufficient amount of previous matrices $A^{n-1}, A^{n-2},...$



I ask this because I was studying the leading term in the matrix $A^n$ where $$A=
begin{bmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
7 & 8 & 9 \
end{bmatrix}
$$

which can be found in a ridiculous closed form (https://oeis.org/A321045), but has the following recurrence relation between the nth powers of the matrix:
$$a(n+2)=15a(n+1)+18a(n)$$
for $n ge 1$. How does one find a relation like this or equivalently find it's corresponding generating function - $frac{3x^2+14x-1}{18x^2+15x-1}$ ?







matrices recurrence-relations generating-functions






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 8 at 23:22









Peter ForemanPeter Foreman

2,243114




2,243114








  • 1




    $begingroup$
    this is just Cayley-Hamilton. In your case, the determinant is zero, so you don't get the full effect, with would be expected to relate $n+3,n+2,n+1,n$
    $endgroup$
    – Will Jagy
    Jan 8 at 23:25














  • 1




    $begingroup$
    this is just Cayley-Hamilton. In your case, the determinant is zero, so you don't get the full effect, with would be expected to relate $n+3,n+2,n+1,n$
    $endgroup$
    – Will Jagy
    Jan 8 at 23:25








1




1




$begingroup$
this is just Cayley-Hamilton. In your case, the determinant is zero, so you don't get the full effect, with would be expected to relate $n+3,n+2,n+1,n$
$endgroup$
– Will Jagy
Jan 8 at 23:25




$begingroup$
this is just Cayley-Hamilton. In your case, the determinant is zero, so you don't get the full effect, with would be expected to relate $n+3,n+2,n+1,n$
$endgroup$
– Will Jagy
Jan 8 at 23:25










1 Answer
1






active

oldest

votes


















2












$begingroup$

First, find some relation between “small” powers of $A$, eg here I guess $A^3=15A^2+18A$ (For this, you can try and test or use Cayley-Hamilton). Then multiply everything by $A^{n-1}$.






share|cite|improve this answer









$endgroup$













    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%2f3066856%2frecurrence-relations-in-matrices-raised-to-natural-powers%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









    2












    $begingroup$

    First, find some relation between “small” powers of $A$, eg here I guess $A^3=15A^2+18A$ (For this, you can try and test or use Cayley-Hamilton). Then multiply everything by $A^{n-1}$.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      First, find some relation between “small” powers of $A$, eg here I guess $A^3=15A^2+18A$ (For this, you can try and test or use Cayley-Hamilton). Then multiply everything by $A^{n-1}$.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        First, find some relation between “small” powers of $A$, eg here I guess $A^3=15A^2+18A$ (For this, you can try and test or use Cayley-Hamilton). Then multiply everything by $A^{n-1}$.






        share|cite|improve this answer









        $endgroup$



        First, find some relation between “small” powers of $A$, eg here I guess $A^3=15A^2+18A$ (For this, you can try and test or use Cayley-Hamilton). Then multiply everything by $A^{n-1}$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 8 at 23:27









        MindlackMindlack

        4,740210




        4,740210






























            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3066856%2frecurrence-relations-in-matrices-raised-to-natural-powers%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

            Human spaceflight

            Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

            File:DeusFollowingSea.jpg