Mixing
Average angle between x and Ax?
$begingroup$
Suppose $Ainmathbb{R}^{ntimes n}$ is a fixed positive definite matrix and $x in mathbb{R}^n$. The cosine of angle between $x$ and $Ax$ is given by
$$
frac{x^T Ax}{|x||Ax|}.
$$
I want to know what the average value of the above expression is when $x$ is uniformly distributed on unit sphere in $mathbb{R}^n$.
Diagonalizing $A$ and some algebraic manipulation, one can show that the answer is equal to
$$
int_{alpha_1^2+cdots+alpha_n^2=1} frac{sum lambda_i alpha_i^2}{sqrt{sumlambda_i^2alpha_i^2}},d alpha_1cdots dalpha_n,
$$
where $lambda_i$, for $1leq ileq n$, are the eigenvalues of $A$. However, I do not know how to calculate the above integral.
linear-algebra spherical-coordinates
asked Jan 16 at 6:43
mathlovermathlover
42
$endgroup$
add a comment |
$begingroup$
Suppose $Ainmathbb{R}^{ntimes n}$ is a fixed positive definite matrix and $x in mathbb{R}^n$. The cosine of angle between $x$ and $Ax$ is given by
$$
frac{x^T Ax}{|x||Ax|}.
$$
I want to know what the average value of the above expression is when $x$ is uniformly distributed on unit sphere in $mathbb{R}^n$.
Diagonalizing $A$ and some algebraic manipulation, one can show that the answer is equal to
$$
int_{alpha_1^2+cdots+alpha_n^2=1} frac{sum lambda_i alpha_i^2}{sqrt{sumlambda_i^2alpha_i^2}},d alpha_1cdots dalpha_n,
$$
where $lambda_i$, for $1leq ileq n$, are the eigenvalues of $A$. However, I do not know how to calculate the above integral.
linear-algebra spherical-coordinates
asked Jan 16 at 6:43
mathlovermathlover
42
$endgroup$
1
$begingroup$
What does it mean: "uniformly distributed in $mathbb{R}^n$"?!
$endgroup$
– metamorphy
Jan 16 at 6:48
3
$begingroup$
"The angle between $x$ and $Ax$" The cosine of the angle between them is given by that expression.
$endgroup$
– Arthur
Jan 16 at 6:49
$begingroup$
Maybe you mean "uniformly distributed on the unit sphere"?
$endgroup$
– GReyes
Jan 16 at 7:38
$begingroup$
You write anything and you ignore comments... Moreover, to obtain the average of the cosinus of angle, you must divide your formula by the measure of $S^{n-1}$.
$endgroup$
– loup blanc
Jan 16 at 19:40
$begingroup$
@GReyes Thanks, I modified it.
$endgroup$
– mathlover
Jan 16 at 21:56
add a comment |
$begingroup$
Suppose $Ainmathbb{R}^{ntimes n}$ is a fixed positive definite matrix and $x in mathbb{R}^n$. The cosine of angle between $x$ and $Ax$ is given by
$$
frac{x^T Ax}{|x||Ax|}.
$$
I want to know what the average value of the above expression is when $x$ is uniformly distributed on unit sphere in $mathbb{R}^n$.
Diagonalizing $A$ and some algebraic manipulation, one can show that the answer is equal to
$$
int_{alpha_1^2+cdots+alpha_n^2=1} frac{sum lambda_i alpha_i^2}{sqrt{sumlambda_i^2alpha_i^2}},d alpha_1cdots dalpha_n,
$$
where $lambda_i$, for $1leq ileq n$, are the eigenvalues of $A$. However, I do not know how to calculate the above integral.
linear-algebra spherical-coordinates
asked Jan 16 at 6:43
mathlovermathlover
42
$endgroup$
Suppose $Ainmathbb{R}^{ntimes n}$ is a fixed positive definite matrix and $x in mathbb{R}^n$. The cosine of angle between $x$ and $Ax$ is given by
$$
frac{x^T Ax}{|x||Ax|}.
$$
I want to know what the average value of the above expression is when $x$ is uniformly distributed on unit sphere in $mathbb{R}^n$.
Diagonalizing $A$ and some algebraic manipulation, one can show that the answer is equal to
$$
int_{alpha_1^2+cdots+alpha_n^2=1} frac{sum lambda_i alpha_i^2}{sqrt{sumlambda_i^2alpha_i^2}},d alpha_1cdots dalpha_n,
$$
where $lambda_i$, for $1leq ileq n$, are the eigenvalues of $A$. However, I do not know how to calculate the above integral.
linear-algebra spherical-coordinates
linear-algebra spherical-coordinates
asked Jan 16 at 6:43
mathlovermathlover
42
asked Jan 16 at 6:43
mathlovermathlover
42
edited Jan 17 at 6:31
mathlover
asked Jan 16 at 6:43
mathlovermathlover
42
asked Jan 16 at 6:43
mathlovermathlover
42
asked Jan 16 at 6:43
mathlovermathlover
42
42
1
$begingroup$
What does it mean: "uniformly distributed in $mathbb{R}^n$"?!
$endgroup$
– metamorphy
Jan 16 at 6:48
3
$begingroup$
"The angle between $x$ and $Ax$" The cosine of the angle between them is given by that expression.
$endgroup$
– Arthur
Jan 16 at 6:49
$begingroup$
Maybe you mean "uniformly distributed on the unit sphere"?
$endgroup$
– GReyes
Jan 16 at 7:38
$begingroup$
You write anything and you ignore comments... Moreover, to obtain the average of the cosinus of angle, you must divide your formula by the measure of $S^{n-1}$.
$endgroup$
– loup blanc
Jan 16 at 19:40
$begingroup$
@GReyes Thanks, I modified it.
$endgroup$
– mathlover
Jan 16 at 21:56
add a comment |
1
$begingroup$
What does it mean: "uniformly distributed in $mathbb{R}^n$"?!
$endgroup$
– metamorphy
Jan 16 at 6:48
3
$begingroup$
"The angle between $x$ and $Ax$" The cosine of the angle between them is given by that expression.
$endgroup$
– Arthur
Jan 16 at 6:49
$begingroup$
Maybe you mean "uniformly distributed on the unit sphere"?
$endgroup$
– GReyes
Jan 16 at 7:38
$begingroup$
You write anything and you ignore comments... Moreover, to obtain the average of the cosinus of angle, you must divide your formula by the measure of $S^{n-1}$.
$endgroup$
– loup blanc
Jan 16 at 19:40
$begingroup$
@GReyes Thanks, I modified it.
$endgroup$
– mathlover
Jan 16 at 21:56
1
1
$begingroup$
What does it mean: "uniformly distributed in $mathbb{R}^n$"?!
$endgroup$
– metamorphy
Jan 16 at 6:48
$begingroup$
What does it mean: "uniformly distributed in $mathbb{R}^n$"?!
$endgroup$
– metamorphy
Jan 16 at 6:48
3
3
$begingroup$
"The angle between $x$ and $Ax$" The cosine of the angle between them is given by that expression.
$endgroup$
– Arthur
Jan 16 at 6:49
$begingroup$
"The angle between $x$ and $Ax$" The cosine of the angle between them is given by that expression.
$endgroup$
– Arthur
Jan 16 at 6:49
$begingroup$
Maybe you mean "uniformly distributed on the unit sphere"?
$endgroup$
– GReyes
Jan 16 at 7:38
$begingroup$
Maybe you mean "uniformly distributed on the unit sphere"?
$endgroup$
– GReyes
Jan 16 at 7:38
$begingroup$
You write anything and you ignore comments... Moreover, to obtain the average of the cosinus of angle, you must divide your formula by the measure of $S^{n-1}$.
$endgroup$
– loup blanc
Jan 16 at 19:40
$begingroup$
You write anything and you ignore comments... Moreover, to obtain the average of the cosinus of angle, you must divide your formula by the measure of $S^{n-1}$.
$endgroup$
– loup blanc
Jan 16 at 19:40
$begingroup$
@GReyes Thanks, I modified it.
$endgroup$
– mathlover
Jan 16 at 21:56
$begingroup$
@GReyes Thanks, I modified it.
$endgroup$
– mathlover
Jan 16 at 21:56
add a comment |
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1
$begingroup$
What does it mean: "uniformly distributed in $mathbb{R}^n$"?!
$endgroup$
– metamorphy
Jan 16 at 6:48
3
$begingroup$
"The angle between $x$ and $Ax$" The cosine of the angle between them is given by that expression.
$endgroup$
– Arthur
Jan 16 at 6:49
$begingroup$
Maybe you mean "uniformly distributed on the unit sphere"?
$endgroup$
– GReyes
Jan 16 at 7:38
$begingroup$
You write anything and you ignore comments... Moreover, to obtain the average of the cosinus of angle, you must divide your formula by the measure of $S^{n-1}$.
$endgroup$
– loup blanc
Jan 16 at 19:40
$begingroup$
@GReyes Thanks, I modified it.
$endgroup$
– mathlover
Jan 16 at 21:56