Positive semi-definite vs positive definite












13












$begingroup$


I am confused about the difference between positive semi-definite and positive definite.



May I understand that positive semi-definite means symmetric and $x'Ax ge 0$, while positive definite means symmetric and $x'Ax gt 0$?










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    yes pretty much.
    $endgroup$
    – jacob smith
    Apr 8 '16 at 18:36










  • $begingroup$
    I fixed some formatting issues, but you could improve the Question by opening with a mention that you are asking about properties of matrices. In any case I added that as a tag.
    $endgroup$
    – hardmath
    Apr 8 '16 at 18:41










  • $begingroup$
    The inequality for positive definite is often given as $x^TAxge agt0$, giving a positive lower bound.
    $endgroup$
    – robjohn
    Apr 8 '16 at 18:42








  • 1




    $begingroup$
    Also: for positive definite, that condition only applies when $x ne 0$.
    $endgroup$
    – John Hughes
    Apr 8 '16 at 18:58






  • 2




    $begingroup$
    @WillJagy: ah, good point. Thanks for the correction.
    $endgroup$
    – robjohn
    Apr 8 '16 at 19:31
















13












$begingroup$


I am confused about the difference between positive semi-definite and positive definite.



May I understand that positive semi-definite means symmetric and $x'Ax ge 0$, while positive definite means symmetric and $x'Ax gt 0$?










share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    yes pretty much.
    $endgroup$
    – jacob smith
    Apr 8 '16 at 18:36










  • $begingroup$
    I fixed some formatting issues, but you could improve the Question by opening with a mention that you are asking about properties of matrices. In any case I added that as a tag.
    $endgroup$
    – hardmath
    Apr 8 '16 at 18:41










  • $begingroup$
    The inequality for positive definite is often given as $x^TAxge agt0$, giving a positive lower bound.
    $endgroup$
    – robjohn
    Apr 8 '16 at 18:42








  • 1




    $begingroup$
    Also: for positive definite, that condition only applies when $x ne 0$.
    $endgroup$
    – John Hughes
    Apr 8 '16 at 18:58






  • 2




    $begingroup$
    @WillJagy: ah, good point. Thanks for the correction.
    $endgroup$
    – robjohn
    Apr 8 '16 at 19:31














13












13








13


3



$begingroup$


I am confused about the difference between positive semi-definite and positive definite.



May I understand that positive semi-definite means symmetric and $x'Ax ge 0$, while positive definite means symmetric and $x'Ax gt 0$?










share|cite|improve this question











$endgroup$




I am confused about the difference between positive semi-definite and positive definite.



May I understand that positive semi-definite means symmetric and $x'Ax ge 0$, while positive definite means symmetric and $x'Ax gt 0$?







matrices positive-definite






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Apr 8 '16 at 18:40









hardmath

28.9k95297




28.9k95297










asked Apr 8 '16 at 18:35









pippppippp

81118




81118








  • 2




    $begingroup$
    yes pretty much.
    $endgroup$
    – jacob smith
    Apr 8 '16 at 18:36










  • $begingroup$
    I fixed some formatting issues, but you could improve the Question by opening with a mention that you are asking about properties of matrices. In any case I added that as a tag.
    $endgroup$
    – hardmath
    Apr 8 '16 at 18:41










  • $begingroup$
    The inequality for positive definite is often given as $x^TAxge agt0$, giving a positive lower bound.
    $endgroup$
    – robjohn
    Apr 8 '16 at 18:42








  • 1




    $begingroup$
    Also: for positive definite, that condition only applies when $x ne 0$.
    $endgroup$
    – John Hughes
    Apr 8 '16 at 18:58






  • 2




    $begingroup$
    @WillJagy: ah, good point. Thanks for the correction.
    $endgroup$
    – robjohn
    Apr 8 '16 at 19:31














  • 2




    $begingroup$
    yes pretty much.
    $endgroup$
    – jacob smith
    Apr 8 '16 at 18:36










  • $begingroup$
    I fixed some formatting issues, but you could improve the Question by opening with a mention that you are asking about properties of matrices. In any case I added that as a tag.
    $endgroup$
    – hardmath
    Apr 8 '16 at 18:41










  • $begingroup$
    The inequality for positive definite is often given as $x^TAxge agt0$, giving a positive lower bound.
    $endgroup$
    – robjohn
    Apr 8 '16 at 18:42








  • 1




    $begingroup$
    Also: for positive definite, that condition only applies when $x ne 0$.
    $endgroup$
    – John Hughes
    Apr 8 '16 at 18:58






  • 2




    $begingroup$
    @WillJagy: ah, good point. Thanks for the correction.
    $endgroup$
    – robjohn
    Apr 8 '16 at 19:31








2




2




$begingroup$
yes pretty much.
$endgroup$
– jacob smith
Apr 8 '16 at 18:36




$begingroup$
yes pretty much.
$endgroup$
– jacob smith
Apr 8 '16 at 18:36












$begingroup$
I fixed some formatting issues, but you could improve the Question by opening with a mention that you are asking about properties of matrices. In any case I added that as a tag.
$endgroup$
– hardmath
Apr 8 '16 at 18:41




$begingroup$
I fixed some formatting issues, but you could improve the Question by opening with a mention that you are asking about properties of matrices. In any case I added that as a tag.
$endgroup$
– hardmath
Apr 8 '16 at 18:41












$begingroup$
The inequality for positive definite is often given as $x^TAxge agt0$, giving a positive lower bound.
$endgroup$
– robjohn
Apr 8 '16 at 18:42






$begingroup$
The inequality for positive definite is often given as $x^TAxge agt0$, giving a positive lower bound.
$endgroup$
– robjohn
Apr 8 '16 at 18:42






1




1




$begingroup$
Also: for positive definite, that condition only applies when $x ne 0$.
$endgroup$
– John Hughes
Apr 8 '16 at 18:58




$begingroup$
Also: for positive definite, that condition only applies when $x ne 0$.
$endgroup$
– John Hughes
Apr 8 '16 at 18:58




2




2




$begingroup$
@WillJagy: ah, good point. Thanks for the correction.
$endgroup$
– robjohn
Apr 8 '16 at 19:31




$begingroup$
@WillJagy: ah, good point. Thanks for the correction.
$endgroup$
– robjohn
Apr 8 '16 at 19:31










3 Answers
3






active

oldest

votes


















4












$begingroup$

Yes. In general a matrix $A$ is called...





  • positive definite if for any vector $x neq 0$, $x' A x > 0$


  • positive semi definite if $x' A x geq 0$.



    • nonnegative definite if it is either positive definite or positive semi definite




  • negative definite if $x' A x < 0$.


  • negative semi definite if $x' A x leq 0$.



    • nonpositive definite if it is either negative definite or negative semi definite




  • indefinite if it is nothing of those.



    Literature: e.g. Harville (1997) Matrix Algebra From A Statisticians's Perspective Section 14.2








share|cite|improve this answer











$endgroup$









  • 3




    $begingroup$
    Aren't positive semidefinite matrices already a superset of positive definite matrices? So nonnegative definite and positive semidefinite are the same.
    $endgroup$
    – Rahul
    Jun 1 '16 at 14:57










  • $begingroup$
    Yes, they are the same; but as you can read the expression from time to time (in the mentioned literatur or in this question), i stated it.
    $endgroup$
    – Qaswed
    Jun 1 '16 at 15:16








  • 2




    $begingroup$
    That's true, but it would be clearer if you combined the two definitions instead of having them in separate bullet points as though they were different. So something like "positive semidefinite or nonnegative definite if $x^T Axge 0$", and similarly for negative semidefinite / nonpositive definite.
    $endgroup$
    – Rahul
    Jun 3 '16 at 7:34



















2












$begingroup$

A great source for results about positive (semi-)definite matrices is Chapter 7 in Horn, Johnson (2013) Matrix Analysis, 2nd edition.
One result I found particularly interesting:




Corollary 7.1.7. A positive semidefinite matrix is positive definite if and only if it is nonsingular.







share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    A symmetric matrix A is said to be positive definite if for for all non zero X $X^tAX>0$ and it said be positive semidefinite if their exist some nonzero X such that $X^tAX>=0$.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      Welcome to MSE. Your answer adds nothing new to the already existing answers.
      $endgroup$
      – José Carlos Santos
      Dec 11 '18 at 14:01






    • 1




      $begingroup$
      Sorry, but I didn't seen the above existing answer.
      $endgroup$
      – MANI SHANKAR PANDEY
      Dec 11 '18 at 14:09











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    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4












    $begingroup$

    Yes. In general a matrix $A$ is called...





    • positive definite if for any vector $x neq 0$, $x' A x > 0$


    • positive semi definite if $x' A x geq 0$.



      • nonnegative definite if it is either positive definite or positive semi definite




    • negative definite if $x' A x < 0$.


    • negative semi definite if $x' A x leq 0$.



      • nonpositive definite if it is either negative definite or negative semi definite




    • indefinite if it is nothing of those.



      Literature: e.g. Harville (1997) Matrix Algebra From A Statisticians's Perspective Section 14.2








    share|cite|improve this answer











    $endgroup$









    • 3




      $begingroup$
      Aren't positive semidefinite matrices already a superset of positive definite matrices? So nonnegative definite and positive semidefinite are the same.
      $endgroup$
      – Rahul
      Jun 1 '16 at 14:57










    • $begingroup$
      Yes, they are the same; but as you can read the expression from time to time (in the mentioned literatur or in this question), i stated it.
      $endgroup$
      – Qaswed
      Jun 1 '16 at 15:16








    • 2




      $begingroup$
      That's true, but it would be clearer if you combined the two definitions instead of having them in separate bullet points as though they were different. So something like "positive semidefinite or nonnegative definite if $x^T Axge 0$", and similarly for negative semidefinite / nonpositive definite.
      $endgroup$
      – Rahul
      Jun 3 '16 at 7:34
















    4












    $begingroup$

    Yes. In general a matrix $A$ is called...





    • positive definite if for any vector $x neq 0$, $x' A x > 0$


    • positive semi definite if $x' A x geq 0$.



      • nonnegative definite if it is either positive definite or positive semi definite




    • negative definite if $x' A x < 0$.


    • negative semi definite if $x' A x leq 0$.



      • nonpositive definite if it is either negative definite or negative semi definite




    • indefinite if it is nothing of those.



      Literature: e.g. Harville (1997) Matrix Algebra From A Statisticians's Perspective Section 14.2








    share|cite|improve this answer











    $endgroup$









    • 3




      $begingroup$
      Aren't positive semidefinite matrices already a superset of positive definite matrices? So nonnegative definite and positive semidefinite are the same.
      $endgroup$
      – Rahul
      Jun 1 '16 at 14:57










    • $begingroup$
      Yes, they are the same; but as you can read the expression from time to time (in the mentioned literatur or in this question), i stated it.
      $endgroup$
      – Qaswed
      Jun 1 '16 at 15:16








    • 2




      $begingroup$
      That's true, but it would be clearer if you combined the two definitions instead of having them in separate bullet points as though they were different. So something like "positive semidefinite or nonnegative definite if $x^T Axge 0$", and similarly for negative semidefinite / nonpositive definite.
      $endgroup$
      – Rahul
      Jun 3 '16 at 7:34














    4












    4








    4





    $begingroup$

    Yes. In general a matrix $A$ is called...





    • positive definite if for any vector $x neq 0$, $x' A x > 0$


    • positive semi definite if $x' A x geq 0$.



      • nonnegative definite if it is either positive definite or positive semi definite




    • negative definite if $x' A x < 0$.


    • negative semi definite if $x' A x leq 0$.



      • nonpositive definite if it is either negative definite or negative semi definite




    • indefinite if it is nothing of those.



      Literature: e.g. Harville (1997) Matrix Algebra From A Statisticians's Perspective Section 14.2








    share|cite|improve this answer











    $endgroup$



    Yes. In general a matrix $A$ is called...





    • positive definite if for any vector $x neq 0$, $x' A x > 0$


    • positive semi definite if $x' A x geq 0$.



      • nonnegative definite if it is either positive definite or positive semi definite




    • negative definite if $x' A x < 0$.


    • negative semi definite if $x' A x leq 0$.



      • nonpositive definite if it is either negative definite or negative semi definite




    • indefinite if it is nothing of those.



      Literature: e.g. Harville (1997) Matrix Algebra From A Statisticians's Perspective Section 14.2









    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 11 '18 at 13:34

























    answered Jun 1 '16 at 14:32









    QaswedQaswed

    301313




    301313








    • 3




      $begingroup$
      Aren't positive semidefinite matrices already a superset of positive definite matrices? So nonnegative definite and positive semidefinite are the same.
      $endgroup$
      – Rahul
      Jun 1 '16 at 14:57










    • $begingroup$
      Yes, they are the same; but as you can read the expression from time to time (in the mentioned literatur or in this question), i stated it.
      $endgroup$
      – Qaswed
      Jun 1 '16 at 15:16








    • 2




      $begingroup$
      That's true, but it would be clearer if you combined the two definitions instead of having them in separate bullet points as though they were different. So something like "positive semidefinite or nonnegative definite if $x^T Axge 0$", and similarly for negative semidefinite / nonpositive definite.
      $endgroup$
      – Rahul
      Jun 3 '16 at 7:34














    • 3




      $begingroup$
      Aren't positive semidefinite matrices already a superset of positive definite matrices? So nonnegative definite and positive semidefinite are the same.
      $endgroup$
      – Rahul
      Jun 1 '16 at 14:57










    • $begingroup$
      Yes, they are the same; but as you can read the expression from time to time (in the mentioned literatur or in this question), i stated it.
      $endgroup$
      – Qaswed
      Jun 1 '16 at 15:16








    • 2




      $begingroup$
      That's true, but it would be clearer if you combined the two definitions instead of having them in separate bullet points as though they were different. So something like "positive semidefinite or nonnegative definite if $x^T Axge 0$", and similarly for negative semidefinite / nonpositive definite.
      $endgroup$
      – Rahul
      Jun 3 '16 at 7:34








    3




    3




    $begingroup$
    Aren't positive semidefinite matrices already a superset of positive definite matrices? So nonnegative definite and positive semidefinite are the same.
    $endgroup$
    – Rahul
    Jun 1 '16 at 14:57




    $begingroup$
    Aren't positive semidefinite matrices already a superset of positive definite matrices? So nonnegative definite and positive semidefinite are the same.
    $endgroup$
    – Rahul
    Jun 1 '16 at 14:57












    $begingroup$
    Yes, they are the same; but as you can read the expression from time to time (in the mentioned literatur or in this question), i stated it.
    $endgroup$
    – Qaswed
    Jun 1 '16 at 15:16






    $begingroup$
    Yes, they are the same; but as you can read the expression from time to time (in the mentioned literatur or in this question), i stated it.
    $endgroup$
    – Qaswed
    Jun 1 '16 at 15:16






    2




    2




    $begingroup$
    That's true, but it would be clearer if you combined the two definitions instead of having them in separate bullet points as though they were different. So something like "positive semidefinite or nonnegative definite if $x^T Axge 0$", and similarly for negative semidefinite / nonpositive definite.
    $endgroup$
    – Rahul
    Jun 3 '16 at 7:34




    $begingroup$
    That's true, but it would be clearer if you combined the two definitions instead of having them in separate bullet points as though they were different. So something like "positive semidefinite or nonnegative definite if $x^T Axge 0$", and similarly for negative semidefinite / nonpositive definite.
    $endgroup$
    – Rahul
    Jun 3 '16 at 7:34











    2












    $begingroup$

    A great source for results about positive (semi-)definite matrices is Chapter 7 in Horn, Johnson (2013) Matrix Analysis, 2nd edition.
    One result I found particularly interesting:




    Corollary 7.1.7. A positive semidefinite matrix is positive definite if and only if it is nonsingular.







    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      A great source for results about positive (semi-)definite matrices is Chapter 7 in Horn, Johnson (2013) Matrix Analysis, 2nd edition.
      One result I found particularly interesting:




      Corollary 7.1.7. A positive semidefinite matrix is positive definite if and only if it is nonsingular.







      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        A great source for results about positive (semi-)definite matrices is Chapter 7 in Horn, Johnson (2013) Matrix Analysis, 2nd edition.
        One result I found particularly interesting:




        Corollary 7.1.7. A positive semidefinite matrix is positive definite if and only if it is nonsingular.







        share|cite|improve this answer









        $endgroup$



        A great source for results about positive (semi-)definite matrices is Chapter 7 in Horn, Johnson (2013) Matrix Analysis, 2nd edition.
        One result I found particularly interesting:




        Corollary 7.1.7. A positive semidefinite matrix is positive definite if and only if it is nonsingular.








        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Feb 22 '17 at 13:07









        Elias StrehleElias Strehle

        326215




        326215























            0












            $begingroup$

            A symmetric matrix A is said to be positive definite if for for all non zero X $X^tAX>0$ and it said be positive semidefinite if their exist some nonzero X such that $X^tAX>=0$.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Welcome to MSE. Your answer adds nothing new to the already existing answers.
              $endgroup$
              – José Carlos Santos
              Dec 11 '18 at 14:01






            • 1




              $begingroup$
              Sorry, but I didn't seen the above existing answer.
              $endgroup$
              – MANI SHANKAR PANDEY
              Dec 11 '18 at 14:09
















            0












            $begingroup$

            A symmetric matrix A is said to be positive definite if for for all non zero X $X^tAX>0$ and it said be positive semidefinite if their exist some nonzero X such that $X^tAX>=0$.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Welcome to MSE. Your answer adds nothing new to the already existing answers.
              $endgroup$
              – José Carlos Santos
              Dec 11 '18 at 14:01






            • 1




              $begingroup$
              Sorry, but I didn't seen the above existing answer.
              $endgroup$
              – MANI SHANKAR PANDEY
              Dec 11 '18 at 14:09














            0












            0








            0





            $begingroup$

            A symmetric matrix A is said to be positive definite if for for all non zero X $X^tAX>0$ and it said be positive semidefinite if their exist some nonzero X such that $X^tAX>=0$.






            share|cite|improve this answer









            $endgroup$



            A symmetric matrix A is said to be positive definite if for for all non zero X $X^tAX>0$ and it said be positive semidefinite if their exist some nonzero X such that $X^tAX>=0$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 11 '18 at 13:42









            MANI SHANKAR PANDEYMANI SHANKAR PANDEY

            547




            547












            • $begingroup$
              Welcome to MSE. Your answer adds nothing new to the already existing answers.
              $endgroup$
              – José Carlos Santos
              Dec 11 '18 at 14:01






            • 1




              $begingroup$
              Sorry, but I didn't seen the above existing answer.
              $endgroup$
              – MANI SHANKAR PANDEY
              Dec 11 '18 at 14:09


















            • $begingroup$
              Welcome to MSE. Your answer adds nothing new to the already existing answers.
              $endgroup$
              – José Carlos Santos
              Dec 11 '18 at 14:01






            • 1




              $begingroup$
              Sorry, but I didn't seen the above existing answer.
              $endgroup$
              – MANI SHANKAR PANDEY
              Dec 11 '18 at 14:09
















            $begingroup$
            Welcome to MSE. Your answer adds nothing new to the already existing answers.
            $endgroup$
            – José Carlos Santos
            Dec 11 '18 at 14:01




            $begingroup$
            Welcome to MSE. Your answer adds nothing new to the already existing answers.
            $endgroup$
            – José Carlos Santos
            Dec 11 '18 at 14:01




            1




            1




            $begingroup$
            Sorry, but I didn't seen the above existing answer.
            $endgroup$
            – MANI SHANKAR PANDEY
            Dec 11 '18 at 14:09




            $begingroup$
            Sorry, but I didn't seen the above existing answer.
            $endgroup$
            – MANI SHANKAR PANDEY
            Dec 11 '18 at 14:09


















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