find vector x so that vector: diag(x).C.x has all components equal, where C is positive-definite
$begingroup$
PROBLEM: I am trying to find closed form solutions or provable general properties of solutions for the solution $x$ of the following:
Find $begin{bmatrix} x_1 \ x_2 \ vdots \ x_N end{bmatrix}$ where:
$$diag(x) C x == begin{bmatrix} 1 \ vdots \ 1 end{bmatrix}$$
...(or more generally equals $begin{bmatrix} t \ vdots \ t end{bmatrix}$ for any positive constant $t>0$).
The solution, $x$, is an $N$-by-1 (column) vector. The right hand side, the column vector of ones, is as well.
$C$ is an $N$-by-$N$ correlation matrix: it is symmetric, the diagonal elements are exactly $1$, and the off-diagonal elements are in the range $-1 < C(j,k) < 1$. Actually for my problem we can assume $0 < C(j,k) < 1$, for all $j ne k$.
We should assume $N>3$ and $C(j,k)$ is distinct for every distinct combination $(j,k)$ where $j ne k$. I have found easy solutions for $N=2$, and where all the off-diagonal elements have the same value.
I am using the convention $M=diag(x)$ to refer to the diagonal square matrix $M$ with whose diagonal is formed by the vector $x$, that is, $M_{j,j} = x_j$.
This problem involves (a) simultaneous polynomial equations, and (b) positive definite matrices (more specifically nonsingular correlation matrices). I am trying to find either a closed form solution (which may not exist) or at least to find if this family of equations has “a name” and some well-known properties within some sub field of algebra. It started out as a signal / data viz problem I got obsessed with after working on it in 2 & 3 variable problems, and realized there wasn't (?) a solution for higher dimensions that was easy/obvious (to me at least).
Anything you got is appreciated.
linear-algebra probability polynomials
edited Jan 17 '14 at 23:40
DanielV
18.2k42855
asked Jan 17 '14 at 18:44
tungsten_carbidetungsten_carbide
215
$endgroup$
add a comment |
$begingroup$
PROBLEM: I am trying to find closed form solutions or provable general properties of solutions for the solution $x$ of the following:
Find $begin{bmatrix} x_1 \ x_2 \ vdots \ x_N end{bmatrix}$ where:
$$diag(x) C x == begin{bmatrix} 1 \ vdots \ 1 end{bmatrix}$$
...(or more generally equals $begin{bmatrix} t \ vdots \ t end{bmatrix}$ for any positive constant $t>0$).
The solution, $x$, is an $N$-by-1 (column) vector. The right hand side, the column vector of ones, is as well.
$C$ is an $N$-by-$N$ correlation matrix: it is symmetric, the diagonal elements are exactly $1$, and the off-diagonal elements are in the range $-1 < C(j,k) < 1$. Actually for my problem we can assume $0 < C(j,k) < 1$, for all $j ne k$.
We should assume $N>3$ and $C(j,k)$ is distinct for every distinct combination $(j,k)$ where $j ne k$. I have found easy solutions for $N=2$, and where all the off-diagonal elements have the same value.
I am using the convention $M=diag(x)$ to refer to the diagonal square matrix $M$ with whose diagonal is formed by the vector $x$, that is, $M_{j,j} = x_j$.
This problem involves (a) simultaneous polynomial equations, and (b) positive definite matrices (more specifically nonsingular correlation matrices). I am trying to find either a closed form solution (which may not exist) or at least to find if this family of equations has “a name” and some well-known properties within some sub field of algebra. It started out as a signal / data viz problem I got obsessed with after working on it in 2 & 3 variable problems, and realized there wasn't (?) a solution for higher dimensions that was easy/obvious (to me at least).
Anything you got is appreciated.
linear-algebra probability polynomials
edited Jan 17 '14 at 23:40
DanielV
18.2k42855
asked Jan 17 '14 at 18:44
tungsten_carbidetungsten_carbide
215
$endgroup$
add a comment |
$begingroup$
PROBLEM: I am trying to find closed form solutions or provable general properties of solutions for the solution $x$ of the following:
Find $begin{bmatrix} x_1 \ x_2 \ vdots \ x_N end{bmatrix}$ where:
$$diag(x) C x == begin{bmatrix} 1 \ vdots \ 1 end{bmatrix}$$
...(or more generally equals $begin{bmatrix} t \ vdots \ t end{bmatrix}$ for any positive constant $t>0$).
The solution, $x$, is an $N$-by-1 (column) vector. The right hand side, the column vector of ones, is as well.
$C$ is an $N$-by-$N$ correlation matrix: it is symmetric, the diagonal elements are exactly $1$, and the off-diagonal elements are in the range $-1 < C(j,k) < 1$. Actually for my problem we can assume $0 < C(j,k) < 1$, for all $j ne k$.
We should assume $N>3$ and $C(j,k)$ is distinct for every distinct combination $(j,k)$ where $j ne k$. I have found easy solutions for $N=2$, and where all the off-diagonal elements have the same value.
I am using the convention $M=diag(x)$ to refer to the diagonal square matrix $M$ with whose diagonal is formed by the vector $x$, that is, $M_{j,j} = x_j$.
This problem involves (a) simultaneous polynomial equations, and (b) positive definite matrices (more specifically nonsingular correlation matrices). I am trying to find either a closed form solution (which may not exist) or at least to find if this family of equations has “a name” and some well-known properties within some sub field of algebra. It started out as a signal / data viz problem I got obsessed with after working on it in 2 & 3 variable problems, and realized there wasn't (?) a solution for higher dimensions that was easy/obvious (to me at least).
Anything you got is appreciated.
linear-algebra probability polynomials
edited Jan 17 '14 at 23:40
DanielV
18.2k42855
asked Jan 17 '14 at 18:44
tungsten_carbidetungsten_carbide
215
$endgroup$
PROBLEM: I am trying to find closed form solutions or provable general properties of solutions for the solution $x$ of the following:
Find $begin{bmatrix} x_1 \ x_2 \ vdots \ x_N end{bmatrix}$ where:
$$diag(x) C x == begin{bmatrix} 1 \ vdots \ 1 end{bmatrix}$$
...(or more generally equals $begin{bmatrix} t \ vdots \ t end{bmatrix}$ for any positive constant $t>0$).
The solution, $x$, is an $N$-by-1 (column) vector. The right hand side, the column vector of ones, is as well.
$C$ is an $N$-by-$N$ correlation matrix: it is symmetric, the diagonal elements are exactly $1$, and the off-diagonal elements are in the range $-1 < C(j,k) < 1$. Actually for my problem we can assume $0 < C(j,k) < 1$, for all $j ne k$.
We should assume $N>3$ and $C(j,k)$ is distinct for every distinct combination $(j,k)$ where $j ne k$. I have found easy solutions for $N=2$, and where all the off-diagonal elements have the same value.
I am using the convention $M=diag(x)$ to refer to the diagonal square matrix $M$ with whose diagonal is formed by the vector $x$, that is, $M_{j,j} = x_j$.
This problem involves (a) simultaneous polynomial equations, and (b) positive definite matrices (more specifically nonsingular correlation matrices). I am trying to find either a closed form solution (which may not exist) or at least to find if this family of equations has “a name” and some well-known properties within some sub field of algebra. It started out as a signal / data viz problem I got obsessed with after working on it in 2 & 3 variable problems, and realized there wasn't (?) a solution for higher dimensions that was easy/obvious (to me at least).
Anything you got is appreciated.
linear-algebra probability polynomials
linear-algebra probability polynomials
edited Jan 17 '14 at 23:40
DanielV
18.2k42855
asked Jan 17 '14 at 18:44
tungsten_carbidetungsten_carbide
215
edited Jan 17 '14 at 23:40
DanielV
18.2k42855
asked Jan 17 '14 at 18:44
tungsten_carbidetungsten_carbide
215
edited Jan 17 '14 at 23:40
DanielV
18.2k42855
edited Jan 17 '14 at 23:40
DanielV
18.2k42855
edited Jan 17 '14 at 23:40
DanielV
18.2k42855
18.2k42855
asked Jan 17 '14 at 18:44
tungsten_carbidetungsten_carbide
215
asked Jan 17 '14 at 18:44
tungsten_carbidetungsten_carbide
215
asked Jan 17 '14 at 18:44
tungsten_carbidetungsten_carbide
215
215
add a comment |
add a comment |
2 Answers
2
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oldest
votes
$begingroup$
No closed form: basically isomorphic to finding a general closed-form solution for the roots of polynomials of arbitrarily high degree. It is however frequently solved by numerical optimization in applications such as the determination of so-called "risk parity" portfolios in finance.
answered Jan 11 at 3:08
tungsten_carbidetungsten_carbide
215
$endgroup$
add a comment |
$begingroup$
I am very much interested in knowing if you made any progress on this question. See my question here: How to solve $mathrm{diag}(x) ; A ; x = mathbf{1}$ for $xinmathbb{R}^n$ with $Ainmathbb{R}^{n times n}$?.
edited Apr 13 '17 at 12:19
Community♦
1
answered Jan 31 '14 at 23:05
Marca85Marca85
484
$endgroup$
add a comment |
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2 Answers
2
active
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2 Answers
2
active
oldest
votes
active
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oldest
votes
$begingroup$
No closed form: basically isomorphic to finding a general closed-form solution for the roots of polynomials of arbitrarily high degree. It is however frequently solved by numerical optimization in applications such as the determination of so-called "risk parity" portfolios in finance.
answered Jan 11 at 3:08
tungsten_carbidetungsten_carbide
215
$endgroup$
add a comment |
$begingroup$
No closed form: basically isomorphic to finding a general closed-form solution for the roots of polynomials of arbitrarily high degree. It is however frequently solved by numerical optimization in applications such as the determination of so-called "risk parity" portfolios in finance.
answered Jan 11 at 3:08
tungsten_carbidetungsten_carbide
215
$endgroup$
add a comment |
$begingroup$
No closed form: basically isomorphic to finding a general closed-form solution for the roots of polynomials of arbitrarily high degree. It is however frequently solved by numerical optimization in applications such as the determination of so-called "risk parity" portfolios in finance.
answered Jan 11 at 3:08
tungsten_carbidetungsten_carbide
215
$endgroup$
No closed form: basically isomorphic to finding a general closed-form solution for the roots of polynomials of arbitrarily high degree. It is however frequently solved by numerical optimization in applications such as the determination of so-called "risk parity" portfolios in finance.
answered Jan 11 at 3:08
tungsten_carbidetungsten_carbide
215
answered Jan 11 at 3:08
tungsten_carbidetungsten_carbide
215
answered Jan 11 at 3:08
tungsten_carbidetungsten_carbide
215
answered Jan 11 at 3:08
tungsten_carbidetungsten_carbide
215
215
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add a comment |
$begingroup$
I am very much interested in knowing if you made any progress on this question. See my question here: How to solve $mathrm{diag}(x) ; A ; x = mathbf{1}$ for $xinmathbb{R}^n$ with $Ainmathbb{R}^{n times n}$?.
edited Apr 13 '17 at 12:19
Community♦
1
answered Jan 31 '14 at 23:05
Marca85Marca85
484
$endgroup$
add a comment |
$begingroup$
I am very much interested in knowing if you made any progress on this question. See my question here: How to solve $mathrm{diag}(x) ; A ; x = mathbf{1}$ for $xinmathbb{R}^n$ with $Ainmathbb{R}^{n times n}$?.
edited Apr 13 '17 at 12:19
Community♦
1
answered Jan 31 '14 at 23:05
Marca85Marca85
484
$endgroup$
add a comment |
$begingroup$
I am very much interested in knowing if you made any progress on this question. See my question here: How to solve $mathrm{diag}(x) ; A ; x = mathbf{1}$ for $xinmathbb{R}^n$ with $Ainmathbb{R}^{n times n}$?.
edited Apr 13 '17 at 12:19
Community♦
1
answered Jan 31 '14 at 23:05
Marca85Marca85
484
$endgroup$
I am very much interested in knowing if you made any progress on this question. See my question here: How to solve $mathrm{diag}(x) ; A ; x = mathbf{1}$ for $xinmathbb{R}^n$ with $Ainmathbb{R}^{n times n}$?.
edited Apr 13 '17 at 12:19
Community♦
1
answered Jan 31 '14 at 23:05
Marca85Marca85
484
edited Apr 13 '17 at 12:19
Community♦
1
edited Apr 13 '17 at 12:19
Community♦
1
edited Apr 13 '17 at 12:19
Community♦
1
1
answered Jan 31 '14 at 23:05
Marca85Marca85
484
answered Jan 31 '14 at 23:05
Marca85Marca85
484
answered Jan 31 '14 at 23:05
Marca85Marca85
484
484
add a comment |
add a comment |
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