If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis...
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
asked Nov 25 at 6:09
MathLover
45710
add a comment |
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
asked Nov 25 at 6:09
MathLover
45710
$<X,Y>=X^TAY$. What should $<cdot,cdot>$ satisfy?
– Yadati Kiran
Nov 25 at 7:01
add a comment |
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
asked Nov 25 at 6:09
MathLover
45710
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
asked Nov 25 at 6:09
MathLover
45710
asked Nov 25 at 6:09
MathLover
45710
asked Nov 25 at 6:09
MathLover
45710
asked Nov 25 at 6:09
MathLover
45710
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
add a comment |
$<X,Y>=X^TAY$. 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
add a comment |
1 Answer
1
active
oldest
votes
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 $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.
answered Nov 25 at 7:08
Yadati Kiran
1,692519
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
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.
answered Nov 25 at 7:08
Yadati Kiran
1,692519
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
add a comment |
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 $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.
answered Nov 25 at 7:08
Yadati Kiran
1,692519
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
add a comment |
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 $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.
answered Nov 25 at 7:08
Yadati Kiran
1,692519
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 $langle alpha X,Yrangle=alphalangle X,Yrangle,quadalphainmathbb{R}^n$.
answered Nov 25 at 7:08
Yadati Kiran
1,692519
answered Nov 25 at 7:08
Yadati Kiran
1,692519
answered Nov 25 at 7:08
Yadati Kiran
1,692519
answered Nov 25 at 7:08
Yadati Kiran
1,692519
1,692519
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
add a comment |
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
add a comment |
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$<X,Y>=X^TAY$. What should $<cdot,cdot>$ satisfy?
– Yadati Kiran
Nov 25 at 7:01