Doubt about the rank-nullity theorem involving a (block) matrix












1












$begingroup$


Let $Pinmathbb{R}^{ntimes n}$ and $Qinmathbb{R}^{ntimes m}$ and define
$$
S:=[Q, PQ, P^2Q, ldots, P^{n-1}Q].
$$

Then $Sinmathbb{R}^{ntimes(mn)}$. Now, the book says that, if $operatorname{rank}(S)<n$, then there exists a unit vector $xiinmathbb{R}^n$ which is orthogonal to each column of $S$ (otherwise $n$ columns of $S$ would be linearly independent and then there would be a non singular minor of $S$ of order $n$).



I do not understand why there exists a unit vector $xiinmathbb{R}^n$ which is orthogonal to each column of $S$. I think that, since $operatorname{rank}(S)<n$, then $operatorname{dim}(operatorname{Ker}(S))=n-operatorname{rank}(S)$ and hence there exists at least one vector $x$ such that $Sx=0$. But such $x$ is in $mathbb{R}^{nm}$ and it is orthogonal to each row of $S$ and it is not a unit vector.



Can someone help me?



Thank You










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Then how about considering $S^{mathrm T}$ instead? If you want the vector to be a unit one, then simply pick $x/vert xvert$.
    $endgroup$
    – xbh
    Dec 9 '18 at 10:26












  • $begingroup$
    @xbh Ok, you're right. By considering $S^T$, by the rank-nullity theorem we get that there exists at least an $xinmathbb{R}^n$ such that $x$ is orthogonal to each row of $S^T$ (that is to each column of $S$). Thank You
    $endgroup$
    – Jeji
    Dec 9 '18 at 10:35










  • $begingroup$
    @Jean-ClaudeArbaut Yes, now it is clear also with the eigenvector. Thank You too
    $endgroup$
    – Jeji
    Dec 9 '18 at 10:35
















1












$begingroup$


Let $Pinmathbb{R}^{ntimes n}$ and $Qinmathbb{R}^{ntimes m}$ and define
$$
S:=[Q, PQ, P^2Q, ldots, P^{n-1}Q].
$$

Then $Sinmathbb{R}^{ntimes(mn)}$. Now, the book says that, if $operatorname{rank}(S)<n$, then there exists a unit vector $xiinmathbb{R}^n$ which is orthogonal to each column of $S$ (otherwise $n$ columns of $S$ would be linearly independent and then there would be a non singular minor of $S$ of order $n$).



I do not understand why there exists a unit vector $xiinmathbb{R}^n$ which is orthogonal to each column of $S$. I think that, since $operatorname{rank}(S)<n$, then $operatorname{dim}(operatorname{Ker}(S))=n-operatorname{rank}(S)$ and hence there exists at least one vector $x$ such that $Sx=0$. But such $x$ is in $mathbb{R}^{nm}$ and it is orthogonal to each row of $S$ and it is not a unit vector.



Can someone help me?



Thank You










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Then how about considering $S^{mathrm T}$ instead? If you want the vector to be a unit one, then simply pick $x/vert xvert$.
    $endgroup$
    – xbh
    Dec 9 '18 at 10:26












  • $begingroup$
    @xbh Ok, you're right. By considering $S^T$, by the rank-nullity theorem we get that there exists at least an $xinmathbb{R}^n$ such that $x$ is orthogonal to each row of $S^T$ (that is to each column of $S$). Thank You
    $endgroup$
    – Jeji
    Dec 9 '18 at 10:35










  • $begingroup$
    @Jean-ClaudeArbaut Yes, now it is clear also with the eigenvector. Thank You too
    $endgroup$
    – Jeji
    Dec 9 '18 at 10:35














1












1








1


0



$begingroup$


Let $Pinmathbb{R}^{ntimes n}$ and $Qinmathbb{R}^{ntimes m}$ and define
$$
S:=[Q, PQ, P^2Q, ldots, P^{n-1}Q].
$$

Then $Sinmathbb{R}^{ntimes(mn)}$. Now, the book says that, if $operatorname{rank}(S)<n$, then there exists a unit vector $xiinmathbb{R}^n$ which is orthogonal to each column of $S$ (otherwise $n$ columns of $S$ would be linearly independent and then there would be a non singular minor of $S$ of order $n$).



I do not understand why there exists a unit vector $xiinmathbb{R}^n$ which is orthogonal to each column of $S$. I think that, since $operatorname{rank}(S)<n$, then $operatorname{dim}(operatorname{Ker}(S))=n-operatorname{rank}(S)$ and hence there exists at least one vector $x$ such that $Sx=0$. But such $x$ is in $mathbb{R}^{nm}$ and it is orthogonal to each row of $S$ and it is not a unit vector.



Can someone help me?



Thank You










share|cite|improve this question











$endgroup$




Let $Pinmathbb{R}^{ntimes n}$ and $Qinmathbb{R}^{ntimes m}$ and define
$$
S:=[Q, PQ, P^2Q, ldots, P^{n-1}Q].
$$

Then $Sinmathbb{R}^{ntimes(mn)}$. Now, the book says that, if $operatorname{rank}(S)<n$, then there exists a unit vector $xiinmathbb{R}^n$ which is orthogonal to each column of $S$ (otherwise $n$ columns of $S$ would be linearly independent and then there would be a non singular minor of $S$ of order $n$).



I do not understand why there exists a unit vector $xiinmathbb{R}^n$ which is orthogonal to each column of $S$. I think that, since $operatorname{rank}(S)<n$, then $operatorname{dim}(operatorname{Ker}(S))=n-operatorname{rank}(S)$ and hence there exists at least one vector $x$ such that $Sx=0$. But such $x$ is in $mathbb{R}^{nm}$ and it is orthogonal to each row of $S$ and it is not a unit vector.



Can someone help me?



Thank You







linear-algebra matrices






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 9 '18 at 10:22







Jeji

















asked Dec 9 '18 at 9:06









JejiJeji

33818




33818








  • 1




    $begingroup$
    Then how about considering $S^{mathrm T}$ instead? If you want the vector to be a unit one, then simply pick $x/vert xvert$.
    $endgroup$
    – xbh
    Dec 9 '18 at 10:26












  • $begingroup$
    @xbh Ok, you're right. By considering $S^T$, by the rank-nullity theorem we get that there exists at least an $xinmathbb{R}^n$ such that $x$ is orthogonal to each row of $S^T$ (that is to each column of $S$). Thank You
    $endgroup$
    – Jeji
    Dec 9 '18 at 10:35










  • $begingroup$
    @Jean-ClaudeArbaut Yes, now it is clear also with the eigenvector. Thank You too
    $endgroup$
    – Jeji
    Dec 9 '18 at 10:35














  • 1




    $begingroup$
    Then how about considering $S^{mathrm T}$ instead? If you want the vector to be a unit one, then simply pick $x/vert xvert$.
    $endgroup$
    – xbh
    Dec 9 '18 at 10:26












  • $begingroup$
    @xbh Ok, you're right. By considering $S^T$, by the rank-nullity theorem we get that there exists at least an $xinmathbb{R}^n$ such that $x$ is orthogonal to each row of $S^T$ (that is to each column of $S$). Thank You
    $endgroup$
    – Jeji
    Dec 9 '18 at 10:35










  • $begingroup$
    @Jean-ClaudeArbaut Yes, now it is clear also with the eigenvector. Thank You too
    $endgroup$
    – Jeji
    Dec 9 '18 at 10:35








1




1




$begingroup$
Then how about considering $S^{mathrm T}$ instead? If you want the vector to be a unit one, then simply pick $x/vert xvert$.
$endgroup$
– xbh
Dec 9 '18 at 10:26






$begingroup$
Then how about considering $S^{mathrm T}$ instead? If you want the vector to be a unit one, then simply pick $x/vert xvert$.
$endgroup$
– xbh
Dec 9 '18 at 10:26














$begingroup$
@xbh Ok, you're right. By considering $S^T$, by the rank-nullity theorem we get that there exists at least an $xinmathbb{R}^n$ such that $x$ is orthogonal to each row of $S^T$ (that is to each column of $S$). Thank You
$endgroup$
– Jeji
Dec 9 '18 at 10:35




$begingroup$
@xbh Ok, you're right. By considering $S^T$, by the rank-nullity theorem we get that there exists at least an $xinmathbb{R}^n$ such that $x$ is orthogonal to each row of $S^T$ (that is to each column of $S$). Thank You
$endgroup$
– Jeji
Dec 9 '18 at 10:35












$begingroup$
@Jean-ClaudeArbaut Yes, now it is clear also with the eigenvector. Thank You too
$endgroup$
– Jeji
Dec 9 '18 at 10:35




$begingroup$
@Jean-ClaudeArbaut Yes, now it is clear also with the eigenvector. Thank You too
$endgroup$
– Jeji
Dec 9 '18 at 10:35










1 Answer
1






active

oldest

votes


















1












$begingroup$

You have $operatorname{rank}(S^{mathrm T}) = operatorname{rank}(S) < n$, so there is an $xin mathbb{R}^n, xneq 0$ and $xinoperatorname{ker}(S^{mathrm T})$, so $x$ is orthogonal to all columns $sin mathbb{R}^n$ of $S$.



$x$ might not be a unit vector (vector of length $1$), but then you can simply take $tilde{x}=frac{x}{||x||}$, which has the same property.






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%2f3032179%2fdoubt-about-the-rank-nullity-theorem-involving-a-block-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









    1












    $begingroup$

    You have $operatorname{rank}(S^{mathrm T}) = operatorname{rank}(S) < n$, so there is an $xin mathbb{R}^n, xneq 0$ and $xinoperatorname{ker}(S^{mathrm T})$, so $x$ is orthogonal to all columns $sin mathbb{R}^n$ of $S$.



    $x$ might not be a unit vector (vector of length $1$), but then you can simply take $tilde{x}=frac{x}{||x||}$, which has the same property.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      You have $operatorname{rank}(S^{mathrm T}) = operatorname{rank}(S) < n$, so there is an $xin mathbb{R}^n, xneq 0$ and $xinoperatorname{ker}(S^{mathrm T})$, so $x$ is orthogonal to all columns $sin mathbb{R}^n$ of $S$.



      $x$ might not be a unit vector (vector of length $1$), but then you can simply take $tilde{x}=frac{x}{||x||}$, which has the same property.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        You have $operatorname{rank}(S^{mathrm T}) = operatorname{rank}(S) < n$, so there is an $xin mathbb{R}^n, xneq 0$ and $xinoperatorname{ker}(S^{mathrm T})$, so $x$ is orthogonal to all columns $sin mathbb{R}^n$ of $S$.



        $x$ might not be a unit vector (vector of length $1$), but then you can simply take $tilde{x}=frac{x}{||x||}$, which has the same property.






        share|cite|improve this answer









        $endgroup$



        You have $operatorname{rank}(S^{mathrm T}) = operatorname{rank}(S) < n$, so there is an $xin mathbb{R}^n, xneq 0$ and $xinoperatorname{ker}(S^{mathrm T})$, so $x$ is orthogonal to all columns $sin mathbb{R}^n$ of $S$.



        $x$ might not be a unit vector (vector of length $1$), but then you can simply take $tilde{x}=frac{x}{||x||}$, which has the same property.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 9 '18 at 10:32









        S. M. RochS. M. Roch

        674315




        674315






























            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%2f3032179%2fdoubt-about-the-rank-nullity-theorem-involving-a-block-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

            Berounka

            Sphinx de Gizeh

            Different font size/position of beamer's navigation symbols template's content depending on regular/plain...