If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis...











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If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $mathbb R^n$




I am studying now bilinear form .I wanted to prove above theorem.



I know that for bilinear for to represent dot product It's matrix is of form $P^TP$
which provide reverse direction.



I not able to prove forword direction.
Any hint will be appreciated










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  • $<X,Y>=X^TAY$. What should $<cdot,cdot>$ satisfy?
    – Yadati Kiran
    Nov 25 at 7:01















up vote
1
down vote

favorite













If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $mathbb R^n$




I am studying now bilinear form .I wanted to prove above theorem.



I know that for bilinear for to represent dot product It's matrix is of form $P^TP$
which provide reverse direction.



I not able to prove forword direction.
Any hint will be appreciated










share|cite|improve this question






















  • $<X,Y>=X^TAY$. What should $<cdot,cdot>$ satisfy?
    – Yadati Kiran
    Nov 25 at 7:01













up vote
1
down vote

favorite









up vote
1
down vote

favorite












If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $mathbb R^n$




I am studying now bilinear form .I wanted to prove above theorem.



I know that for bilinear for to represent dot product It's matrix is of form $P^TP$
which provide reverse direction.



I not able to prove forword direction.
Any hint will be appreciated










share|cite|improve this question














If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $mathbb R^n$




I am studying now bilinear form .I wanted to prove above theorem.



I know that for bilinear for to represent dot product It's matrix is of form $P^TP$
which provide reverse direction.



I not able to prove forword direction.
Any hint will be appreciated







linear-algebra positive-definite bilinear-form symmetric-matrices






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asked Nov 25 at 6:09









MathLover

4129




4129












  • $<X,Y>=X^TAY$. What should $<cdot,cdot>$ satisfy?
    – Yadati Kiran
    Nov 25 at 7:01


















  • $<X,Y>=X^TAY$. What should $<cdot,cdot>$ satisfy?
    – Yadati Kiran
    Nov 25 at 7:01
















$<X,Y>=X^TAY$. What should $<cdot,cdot>$ satisfy?
– Yadati Kiran
Nov 25 at 7:01




$<X,Y>=X^TAY$. What should $<cdot,cdot>$ satisfy?
– Yadati Kiran
Nov 25 at 7:01










1 Answer
1






active

oldest

votes

















up vote
1
down vote













Hint: You must show





  • $langle X,Xranglegeq 0$ and $langle X,Xrangle= 0 iff X=0$

  • $langle X,Yrangle=langle Y,Xrangle$


  • $langle X+Y,Z rangle=langle X,Zrangle+langle Y,Xrangle$ and $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.






share|cite|improve this answer





















  • As an exercise you can try "If the dot product in $mathbb{R}^n$ is defined as $langle X,Yrangle=X^TAY$, then A is symmetric and positive definite.$ "
    – Yadati Kiran
    Nov 25 at 15:48













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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote













Hint: You must show





  • $langle X,Xranglegeq 0$ and $langle X,Xrangle= 0 iff X=0$

  • $langle X,Yrangle=langle Y,Xrangle$


  • $langle X+Y,Z rangle=langle X,Zrangle+langle Y,Xrangle$ and $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.






share|cite|improve this answer





















  • As an exercise you can try "If the dot product in $mathbb{R}^n$ is defined as $langle X,Yrangle=X^TAY$, then A is symmetric and positive definite.$ "
    – Yadati Kiran
    Nov 25 at 15:48

















up vote
1
down vote













Hint: You must show





  • $langle X,Xranglegeq 0$ and $langle X,Xrangle= 0 iff X=0$

  • $langle X,Yrangle=langle Y,Xrangle$


  • $langle X+Y,Z rangle=langle X,Zrangle+langle Y,Xrangle$ and $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.






share|cite|improve this answer





















  • As an exercise you can try "If the dot product in $mathbb{R}^n$ is defined as $langle X,Yrangle=X^TAY$, then A is symmetric and positive definite.$ "
    – Yadati Kiran
    Nov 25 at 15:48















up vote
1
down vote










up vote
1
down vote









Hint: You must show





  • $langle X,Xranglegeq 0$ and $langle X,Xrangle= 0 iff X=0$

  • $langle X,Yrangle=langle Y,Xrangle$


  • $langle X+Y,Z rangle=langle X,Zrangle+langle Y,Xrangle$ and $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.






share|cite|improve this answer












Hint: You must show





  • $langle X,Xranglegeq 0$ and $langle X,Xrangle= 0 iff X=0$

  • $langle X,Yrangle=langle Y,Xrangle$


  • $langle X+Y,Z rangle=langle X,Zrangle+langle Y,Xrangle$ and $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 25 at 7:08









Yadati Kiran

1,245417




1,245417












  • As an exercise you can try "If the dot product in $mathbb{R}^n$ is defined as $langle X,Yrangle=X^TAY$, then A is symmetric and positive definite.$ "
    – Yadati Kiran
    Nov 25 at 15:48




















  • As an exercise you can try "If the dot product in $mathbb{R}^n$ is defined as $langle X,Yrangle=X^TAY$, then A is symmetric and positive definite.$ "
    – Yadati Kiran
    Nov 25 at 15:48


















As an exercise you can try "If the dot product in $mathbb{R}^n$ is defined as $langle X,Yrangle=X^TAY$, then A is symmetric and positive definite.$ "
– Yadati Kiran
Nov 25 at 15:48






As an exercise you can try "If the dot product in $mathbb{R}^n$ is defined as $langle X,Yrangle=X^TAY$, then A is symmetric and positive definite.$ "
– Yadati Kiran
Nov 25 at 15:48




















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