What is a principal minor of a matrix?












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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







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  • 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


















0












$begingroup$


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
















0












0








0





$begingroup$


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













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asked Dec 4 '18 at 18:14









CosmicCosmic

7310




7310








  • 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






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$begingroup$

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












    $begingroup$

    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









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      2 Answers
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      2 Answers
      2






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      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

          321k23210462




          321k23210462























              1












              $begingroup$

              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












                $begingroup$

                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












                  1








                  1





                  $begingroup$

                  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

                  7310




                  7310






























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