Positive semi-definite vs positive definite
$begingroup$
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 $x'Ax gt 0$?
matrices positive-definite
$endgroup$
|
show 1 more comment
$begingroup$
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 $x'Ax gt 0$?
matrices positive-definite
$endgroup$
2
$begingroup$
yes pretty much.
$endgroup$
– jacob smith
Apr 8 '16 at 18:36
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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.
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– 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.
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– 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
|
show 1 more comment
$begingroup$
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 $x'Ax gt 0$?
matrices positive-definite
$endgroup$
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
matrices positive-definite
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
|
show 1 more comment
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
|
show 1 more comment
3 Answers
3
active
oldest
votes
$begingroup$
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
$endgroup$
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
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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 $X^tAX>=0$.
$endgroup$
$begingroup$
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
add a comment |
Your Answer
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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
$endgroup$
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
add a comment |
$begingroup$
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
$endgroup$
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
add a comment |
$begingroup$
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
$endgroup$
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
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
add a comment |
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
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
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.
answered Feb 22 '17 at 13:07
Elias StrehleElias Strehle
326215
326215
add a comment |
add a comment |
$begingroup$
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 $X^tAX>=0$.
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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 $X^tAX>=0$.
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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 $X^tAX>=0$.
$endgroup$
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 $X^tAX>=0$.
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
add a comment |
$begingroup$
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
add a comment |
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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