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












1















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







share|cite|improve this question






















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
















1















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







share|cite|improve this question






















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














1












1








1








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







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




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 25 at 6:09









MathLover

45710




45710












  • $<X,Y>=X^TAY$. 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


















1














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
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active

oldest

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1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1














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


















1














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
















1












1








1






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

1,694519




1,694519












  • 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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