If $A$ is a positive definite matrix and $textdetB=0$, then $textdet(A+B)$ is positive?

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If $A$ is a positive definite matrix and a symmetric matrix $B$ satisfies $textdetB=0$,
then $textdet(A+B)$ is positive?




I tired to find a counterexample for this, but I couldn't.










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  • 1




    Not sure.. What about A = diag(1,1), B= diag(-1,0)?
    – James
    Sep 6 at 11:19














up vote
0
down vote

favorite













If $A$ is a positive definite matrix and a symmetric matrix $B$ satisfies $textdetB=0$,
then $textdet(A+B)$ is positive?




I tired to find a counterexample for this, but I couldn't.










share|cite|improve this question



















  • 1




    Not sure.. What about A = diag(1,1), B= diag(-1,0)?
    – James
    Sep 6 at 11:19












up vote
0
down vote

favorite









up vote
0
down vote

favorite












If $A$ is a positive definite matrix and a symmetric matrix $B$ satisfies $textdetB=0$,
then $textdet(A+B)$ is positive?




I tired to find a counterexample for this, but I couldn't.










share|cite|improve this question
















If $A$ is a positive definite matrix and a symmetric matrix $B$ satisfies $textdetB=0$,
then $textdet(A+B)$ is positive?




I tired to find a counterexample for this, but I couldn't.







determinant positive-definite






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share|cite|improve this question













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edited Sep 6 at 11:11

























asked Sep 6 at 11:03









kayak

576318




576318







  • 1




    Not sure.. What about A = diag(1,1), B= diag(-1,0)?
    – James
    Sep 6 at 11:19












  • 1




    Not sure.. What about A = diag(1,1), B= diag(-1,0)?
    – James
    Sep 6 at 11:19







1




1




Not sure.. What about A = diag(1,1), B= diag(-1,0)?
– James
Sep 6 at 11:19




Not sure.. What about A = diag(1,1), B= diag(-1,0)?
– James
Sep 6 at 11:19










1 Answer
1






active

oldest

votes

















up vote
0
down vote



accepted










Counterexample:




$$mathbf A=beginpmatrix-2&1\1&-7endpmatrix\mathbf B=beginpmatrix1&3\3&9endpmatrix$$ $det(mathbfA)>0$ and $det(mathbfB)=0$, but $det(mathbfA+mathbfB)=(-1)cdot 2-4cdot 4=-18<0$.







share|cite|improve this answer






















  • Thanks. Umm actually $B$ was also a symmetric matrix. Is that okay to change the question?
    – kayak
    Sep 6 at 11:11










  • @kayak Sure! I'll delete my answer then, and think some more.
    – cansomeonehelpmeout
    Sep 6 at 11:12






  • 1




    As a guide for the future: Editing the question after someone's provided an answer is generally frowned upon, for it makes the original answer look silly, when the real silliness was in being sloppy about asking the question.
    – John Hughes
    Sep 6 at 12:03










  • @JohnHughes Oh sorry. Ill keep that in mind.
    – kayak
    Sep 7 at 4:25










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote



accepted










Counterexample:




$$mathbf A=beginpmatrix-2&1\1&-7endpmatrix\mathbf B=beginpmatrix1&3\3&9endpmatrix$$ $det(mathbfA)>0$ and $det(mathbfB)=0$, but $det(mathbfA+mathbfB)=(-1)cdot 2-4cdot 4=-18<0$.







share|cite|improve this answer






















  • Thanks. Umm actually $B$ was also a symmetric matrix. Is that okay to change the question?
    – kayak
    Sep 6 at 11:11










  • @kayak Sure! I'll delete my answer then, and think some more.
    – cansomeonehelpmeout
    Sep 6 at 11:12






  • 1




    As a guide for the future: Editing the question after someone's provided an answer is generally frowned upon, for it makes the original answer look silly, when the real silliness was in being sloppy about asking the question.
    – John Hughes
    Sep 6 at 12:03










  • @JohnHughes Oh sorry. Ill keep that in mind.
    – kayak
    Sep 7 at 4:25














up vote
0
down vote



accepted










Counterexample:




$$mathbf A=beginpmatrix-2&1\1&-7endpmatrix\mathbf B=beginpmatrix1&3\3&9endpmatrix$$ $det(mathbfA)>0$ and $det(mathbfB)=0$, but $det(mathbfA+mathbfB)=(-1)cdot 2-4cdot 4=-18<0$.







share|cite|improve this answer






















  • Thanks. Umm actually $B$ was also a symmetric matrix. Is that okay to change the question?
    – kayak
    Sep 6 at 11:11










  • @kayak Sure! I'll delete my answer then, and think some more.
    – cansomeonehelpmeout
    Sep 6 at 11:12






  • 1




    As a guide for the future: Editing the question after someone's provided an answer is generally frowned upon, for it makes the original answer look silly, when the real silliness was in being sloppy about asking the question.
    – John Hughes
    Sep 6 at 12:03










  • @JohnHughes Oh sorry. Ill keep that in mind.
    – kayak
    Sep 7 at 4:25












up vote
0
down vote



accepted







up vote
0
down vote



accepted






Counterexample:




$$mathbf A=beginpmatrix-2&1\1&-7endpmatrix\mathbf B=beginpmatrix1&3\3&9endpmatrix$$ $det(mathbfA)>0$ and $det(mathbfB)=0$, but $det(mathbfA+mathbfB)=(-1)cdot 2-4cdot 4=-18<0$.







share|cite|improve this answer














Counterexample:




$$mathbf A=beginpmatrix-2&1\1&-7endpmatrix\mathbf B=beginpmatrix1&3\3&9endpmatrix$$ $det(mathbfA)>0$ and $det(mathbfB)=0$, but $det(mathbfA+mathbfB)=(-1)cdot 2-4cdot 4=-18<0$.








share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Sep 6 at 11:26

























answered Sep 6 at 11:08









cansomeonehelpmeout

5,5573830




5,5573830











  • Thanks. Umm actually $B$ was also a symmetric matrix. Is that okay to change the question?
    – kayak
    Sep 6 at 11:11










  • @kayak Sure! I'll delete my answer then, and think some more.
    – cansomeonehelpmeout
    Sep 6 at 11:12






  • 1




    As a guide for the future: Editing the question after someone's provided an answer is generally frowned upon, for it makes the original answer look silly, when the real silliness was in being sloppy about asking the question.
    – John Hughes
    Sep 6 at 12:03










  • @JohnHughes Oh sorry. Ill keep that in mind.
    – kayak
    Sep 7 at 4:25
















  • Thanks. Umm actually $B$ was also a symmetric matrix. Is that okay to change the question?
    – kayak
    Sep 6 at 11:11










  • @kayak Sure! I'll delete my answer then, and think some more.
    – cansomeonehelpmeout
    Sep 6 at 11:12






  • 1




    As a guide for the future: Editing the question after someone's provided an answer is generally frowned upon, for it makes the original answer look silly, when the real silliness was in being sloppy about asking the question.
    – John Hughes
    Sep 6 at 12:03










  • @JohnHughes Oh sorry. Ill keep that in mind.
    – kayak
    Sep 7 at 4:25















Thanks. Umm actually $B$ was also a symmetric matrix. Is that okay to change the question?
– kayak
Sep 6 at 11:11




Thanks. Umm actually $B$ was also a symmetric matrix. Is that okay to change the question?
– kayak
Sep 6 at 11:11












@kayak Sure! I'll delete my answer then, and think some more.
– cansomeonehelpmeout
Sep 6 at 11:12




@kayak Sure! I'll delete my answer then, and think some more.
– cansomeonehelpmeout
Sep 6 at 11:12




1




1




As a guide for the future: Editing the question after someone's provided an answer is generally frowned upon, for it makes the original answer look silly, when the real silliness was in being sloppy about asking the question.
– John Hughes
Sep 6 at 12:03




As a guide for the future: Editing the question after someone's provided an answer is generally frowned upon, for it makes the original answer look silly, when the real silliness was in being sloppy about asking the question.
– John Hughes
Sep 6 at 12:03












@JohnHughes Oh sorry. Ill keep that in mind.
– kayak
Sep 7 at 4:25




@JohnHughes Oh sorry. Ill keep that in mind.
– kayak
Sep 7 at 4:25

















 

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