What is a principal minor of a matrix?












1












$begingroup$


I was going through the book on operation research by Hamdy A.Taha. It referred to principal minor of a hessian matrix. Can someone explain what is meant by a principal minor? Is it different from 'minor of a matrix'?






matrices optimization operations-research







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    A minor is a square submatrix formed by omitting some rows and the same number of columns. A principle minor omits one row and one column.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:15










  • $begingroup$
    Thanks a lot..!
    $endgroup$
    – Cosmic
    Dec 4 '18 at 18:17










  • $begingroup$
    You're welcome. You can delete the question or answer it yourself - don't let it sit on the unanswered queue gaining attention.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:19






  • 3




    $begingroup$
    @EthanBolker That's a "first minor". A principal minor is where the indices of the omitted rows and columns are the same.
    $endgroup$
    – Robert Israel
    Dec 4 '18 at 18:45










  • $begingroup$
    @RobertIsrael Thank you! I think I knew but didn't say that the row and column indices must match. "First minor" is a new term for me. Perhaps you should post this as an answer.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:47


















1












$begingroup$


I was going through the book on operation research by Hamdy A.Taha. It referred to principal minor of a hessian matrix. Can someone explain what is meant by a principal minor? Is it different from 'minor of a matrix'?






matrices optimization operations-research







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    A minor is a square submatrix formed by omitting some rows and the same number of columns. A principle minor omits one row and one column.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:15










  • $begingroup$
    Thanks a lot..!
    $endgroup$
    – Cosmic
    Dec 4 '18 at 18:17










  • $begingroup$
    You're welcome. You can delete the question or answer it yourself - don't let it sit on the unanswered queue gaining attention.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:19






  • 3




    $begingroup$
    @EthanBolker That's a "first minor". A principal minor is where the indices of the omitted rows and columns are the same.
    $endgroup$
    – Robert Israel
    Dec 4 '18 at 18:45










  • $begingroup$
    @RobertIsrael Thank you! I think I knew but didn't say that the row and column indices must match. "First minor" is a new term for me. Perhaps you should post this as an answer.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:47
















1












1








1





$begingroup$


I was going through the book on operation research by Hamdy A.Taha. It referred to principal minor of a hessian matrix. Can someone explain what is meant by a principal minor? Is it different from 'minor of a matrix'?






matrices optimization operations-research







share|cite|improve this question









$endgroup$




I was going through the book on operation research by Hamdy A.Taha. It referred to principal minor of a hessian matrix. Can someone explain what is meant by a principal minor? Is it different from 'minor of a matrix'?






matrices optimization operations-research




matrices optimization operations-research






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 4 '18 at 18:14









CosmicCosmic

8810




8810








  • 1




    $begingroup$
    A minor is a square submatrix formed by omitting some rows and the same number of columns. A principle minor omits one row and one column.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:15










  • $begingroup$
    Thanks a lot..!
    $endgroup$
    – Cosmic
    Dec 4 '18 at 18:17










  • $begingroup$
    You're welcome. You can delete the question or answer it yourself - don't let it sit on the unanswered queue gaining attention.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:19






  • 3




    $begingroup$
    @EthanBolker That's a "first minor". A principal minor is where the indices of the omitted rows and columns are the same.
    $endgroup$
    – Robert Israel
    Dec 4 '18 at 18:45










  • $begingroup$
    @RobertIsrael Thank you! I think I knew but didn't say that the row and column indices must match. "First minor" is a new term for me. Perhaps you should post this as an answer.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:47
















  • 1




    $begingroup$
    A minor is a square submatrix formed by omitting some rows and the same number of columns. A principle minor omits one row and one column.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:15










  • $begingroup$
    Thanks a lot..!
    $endgroup$
    – Cosmic
    Dec 4 '18 at 18:17










  • $begingroup$
    You're welcome. You can delete the question or answer it yourself - don't let it sit on the unanswered queue gaining attention.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:19






  • 3




    $begingroup$
    @EthanBolker That's a "first minor". A principal minor is where the indices of the omitted rows and columns are the same.
    $endgroup$
    – Robert Israel
    Dec 4 '18 at 18:45










  • $begingroup$
    @RobertIsrael Thank you! I think I knew but didn't say that the row and column indices must match. "First minor" is a new term for me. Perhaps you should post this as an answer.
    $endgroup$
    – Ethan Bolker
    Dec 4 '18 at 18:47










1




1




$begingroup$
A minor is a square submatrix formed by omitting some rows and the same number of columns. A principle minor omits one row and one column.
$endgroup$
– Ethan Bolker
Dec 4 '18 at 18:15




$begingroup$
A minor is a square submatrix formed by omitting some rows and the same number of columns. A principle minor omits one row and one column.
$endgroup$
– Ethan Bolker
Dec 4 '18 at 18:15












$begingroup$
Thanks a lot..!
$endgroup$
– Cosmic
Dec 4 '18 at 18:17




$begingroup$
Thanks a lot..!
$endgroup$
– Cosmic
Dec 4 '18 at 18:17












$begingroup$
You're welcome. You can delete the question or answer it yourself - don't let it sit on the unanswered queue gaining attention.
$endgroup$
– Ethan Bolker
Dec 4 '18 at 18:19




$begingroup$
You're welcome. You can delete the question or answer it yourself - don't let it sit on the unanswered queue gaining attention.
$endgroup$
– Ethan Bolker
Dec 4 '18 at 18:19




3




3




$begingroup$
@EthanBolker That's a "first minor". A principal minor is where the indices of the omitted rows and columns are the same.
$endgroup$
– Robert Israel
Dec 4 '18 at 18:45




$begingroup$
@EthanBolker That's a "first minor". A principal minor is where the indices of the omitted rows and columns are the same.
$endgroup$
– Robert Israel
Dec 4 '18 at 18:45












$begingroup$
@RobertIsrael Thank you! I think I knew but didn't say that the row and column indices must match. "First minor" is a new term for me. Perhaps you should post this as an answer.
$endgroup$
– Ethan Bolker
Dec 4 '18 at 18:47






$begingroup$
@RobertIsrael Thank you! I think I knew but didn't say that the row and column indices must match. "First minor" is a new term for me. Perhaps you should post this as an answer.
$endgroup$
– Ethan Bolker
Dec 4 '18 at 18:47












2 Answers
2






active

oldest

votes


















2












$begingroup$

As pointed out by @RobertIsrael, the principal minor is a minor in which the indices of the omitted row and column match.
for example for a $3*3$ matrix:
a principal minor can be created by omitting '1st row and 1st column',
or by omitting '1st row, 2nd row, 1 column, 2nd column' and so on.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    A minor of a matrix $A$ is the determinant of a submatrix formed from $A$ by deleting some (possibly empty,) set of rows and some (possibly empty) set of columns. In order for this determinant to exist, the number of remaining rows must be equal to the number of remaining columns (and there must be at least one remaining row and column). A principal minor of a square matrix is one where the indices of the deleted rows are the same as the indices of the deleted columns.



    Thus for a $3 times 3$ matrix $A$, you could delete nothing (resulting in the determinant of the matrix itself), delete one row and the corresponding column (resulting in one of three possible $2 times 2$
    determinants), or delete two rows and the corresponding two columns
    (resulting in one of the three diagonal elements).






    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%2f3025920%2fwhat-is-a-principal-minor-of-a-matrix%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      As pointed out by @RobertIsrael, the principal minor is a minor in which the indices of the omitted row and column match.
      for example for a $3*3$ matrix:
      a principal minor can be created by omitting '1st row and 1st column',
      or by omitting '1st row, 2nd row, 1 column, 2nd column' and so on.






      share|cite|improve this answer









      $endgroup$


















        2












        $begingroup$

        As pointed out by @RobertIsrael, the principal minor is a minor in which the indices of the omitted row and column match.
        for example for a $3*3$ matrix:
        a principal minor can be created by omitting '1st row and 1st column',
        or by omitting '1st row, 2nd row, 1 column, 2nd column' and so on.






        share|cite|improve this answer









        $endgroup$
















          2












          2








          2





          $begingroup$

          As pointed out by @RobertIsrael, the principal minor is a minor in which the indices of the omitted row and column match.
          for example for a $3*3$ matrix:
          a principal minor can be created by omitting '1st row and 1st column',
          or by omitting '1st row, 2nd row, 1 column, 2nd column' and so on.






          share|cite|improve this answer









          $endgroup$



          As pointed out by @RobertIsrael, the principal minor is a minor in which the indices of the omitted row and column match.
          for example for a $3*3$ matrix:
          a principal minor can be created by omitting '1st row and 1st column',
          or by omitting '1st row, 2nd row, 1 column, 2nd column' and so on.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 4 '18 at 19:14









          CosmicCosmic

          8810




          8810























              1












              $begingroup$

              A minor of a matrix $A$ is the determinant of a submatrix formed from $A$ by deleting some (possibly empty,) set of rows and some (possibly empty) set of columns. In order for this determinant to exist, the number of remaining rows must be equal to the number of remaining columns (and there must be at least one remaining row and column). A principal minor of a square matrix is one where the indices of the deleted rows are the same as the indices of the deleted columns.



              Thus for a $3 times 3$ matrix $A$, you could delete nothing (resulting in the determinant of the matrix itself), delete one row and the corresponding column (resulting in one of three possible $2 times 2$
              determinants), or delete two rows and the corresponding two columns
              (resulting in one of the three diagonal elements).






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                A minor of a matrix $A$ is the determinant of a submatrix formed from $A$ by deleting some (possibly empty,) set of rows and some (possibly empty) set of columns. In order for this determinant to exist, the number of remaining rows must be equal to the number of remaining columns (and there must be at least one remaining row and column). A principal minor of a square matrix is one where the indices of the deleted rows are the same as the indices of the deleted columns.



                Thus for a $3 times 3$ matrix $A$, you could delete nothing (resulting in the determinant of the matrix itself), delete one row and the corresponding column (resulting in one of three possible $2 times 2$
                determinants), or delete two rows and the corresponding two columns
                (resulting in one of the three diagonal elements).






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  A minor of a matrix $A$ is the determinant of a submatrix formed from $A$ by deleting some (possibly empty,) set of rows and some (possibly empty) set of columns. In order for this determinant to exist, the number of remaining rows must be equal to the number of remaining columns (and there must be at least one remaining row and column). A principal minor of a square matrix is one where the indices of the deleted rows are the same as the indices of the deleted columns.



                  Thus for a $3 times 3$ matrix $A$, you could delete nothing (resulting in the determinant of the matrix itself), delete one row and the corresponding column (resulting in one of three possible $2 times 2$
                  determinants), or delete two rows and the corresponding two columns
                  (resulting in one of the three diagonal elements).






                  share|cite|improve this answer









                  $endgroup$



                  A minor of a matrix $A$ is the determinant of a submatrix formed from $A$ by deleting some (possibly empty,) set of rows and some (possibly empty) set of columns. In order for this determinant to exist, the number of remaining rows must be equal to the number of remaining columns (and there must be at least one remaining row and column). A principal minor of a square matrix is one where the indices of the deleted rows are the same as the indices of the deleted columns.



                  Thus for a $3 times 3$ matrix $A$, you could delete nothing (resulting in the determinant of the matrix itself), delete one row and the corresponding column (resulting in one of three possible $2 times 2$
                  determinants), or delete two rows and the corresponding two columns
                  (resulting in one of the three diagonal elements).







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 4 '18 at 20:24









                  Robert IsraelRobert Israel

                  325k23214468




                  325k23214468






























                      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%2f3025920%2fwhat-is-a-principal-minor-of-a-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

                      mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924

                      Image
                      0 I just uploaded a new plugin to my wordpress site and I got this warning: mysqli_query(): Empty query in /home/lucindabrummitt/public_html/blog/wp-includes/wp-db.php on line 1924 Usually if I have a problem, I can find answers on google but this one seems pretty particular and I'm not finding any info on it except for this forum. I'm not terribly techie but if someone can tell me how to read this and how to change it I can follow directions pretty well. Thanks All! PS If I don't have the proper tags here could someone add the appropriate tags? php mysql wordpress share | improve this question asked Nov 16 '18 at 20:41 ...
                      Read more

                      How to change which sound is reproduced for terminal bell?

                      Image
                      3 Recently upgrade to 18.10 from 18.04, and using the terminal is becoming annoying: the new alert sound is a drip ( /usr/share/sounds/gnome/default/alerts/drip.ogg ), and the previous one is still located at ( /usr/share/sounds/ubuntu/stereo/bell.ogg ). My issue is that at the gnome-terminal, when autocompleting or pressing left BOTH sounds are reproduced, at the same time. I want to just have ubuntu/stereo/bell.ogg active, not both . At the moment I have been only able to reduce the volume of the alerts (but still both are played), or turn the alerts down (but I dislike both "solutions"). gnome-terminal 18.10 teminal-bell share | improve this question ...
                      Read more

                      Can I use Tabulator js library in my java Spring + Thymeleaf project?

                      Image
                      0 I want to add Tabulator js to my java project. I added tabulator.min.js, tabulator.min.css to resources folder. After that added to my html page these dependencies as recommending in readme.md. <link rel="stylesheet" th:href="@{/css/tabulator.min.css}"> <script type="text/javascript" th:src="@{/js/tabulator.min.js}"></script> Now my tables view all elements, that I have in the html page, but not correct. In the documentation of Tabulator is example how to create header and table, var table = new Tabulator("#example-table", { columns:[ {title:"Name", field:"name", sorter:"string", width:200, editor:true}, {title:"Age", field:"age", sorter:"...
                      Read more