Find a permutation making the sum of products to equal a given value
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Suppose we have a sequence of integers ${x_i}$ and a permutation of this sequence changing its order ${x_{j_i}}$. We want to find a such a permutation making $sum_i x_ix_{j_i} =C$, where C is a already given value. For example, given $x={1,2,3,4}$ and $C=28$, then the permutation ${2,1,4,3}$ satisfies the above conditions.
If the above is not clear enough, essentially, the question is
Given a column vector $x$ of integer elements, and a constant $C$, find a permutation matrix $A$ such that $x^T A x=C$, where T denotes transpose.
$A$ is certainly not unique, so I am looking for a way to find all solutions.
I try to make $x$ into a square matrix by right multiply it by a row vector, but obviously this will certainly give a singular matrix without an inverse, so I make no progresses.
linear-algebra matrices discrete-mathematics permutations
edited Nov 15 at 4:44
gt6989b
32k22351
asked Nov 15 at 4:05
Ma Joad
799216
add a comment |
up vote
1
down vote
favorite
Suppose we have a sequence of integers ${x_i}$ and a permutation of this sequence changing its order ${x_{j_i}}$. We want to find a such a permutation making $sum_i x_ix_{j_i} =C$, where C is a already given value. For example, given $x={1,2,3,4}$ and $C=28$, then the permutation ${2,1,4,3}$ satisfies the above conditions.
If the above is not clear enough, essentially, the question is
Given a column vector $x$ of integer elements, and a constant $C$, find a permutation matrix $A$ such that $x^T A x=C$, where T denotes transpose.
$A$ is certainly not unique, so I am looking for a way to find all solutions.
I try to make $x$ into a square matrix by right multiply it by a row vector, but obviously this will certainly give a singular matrix without an inverse, so I make no progresses.
linear-algebra matrices discrete-mathematics permutations
edited Nov 15 at 4:44
gt6989b
32k22351
asked Nov 15 at 4:05
Ma Joad
799216
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Suppose we have a sequence of integers ${x_i}$ and a permutation of this sequence changing its order ${x_{j_i}}$. We want to find a such a permutation making $sum_i x_ix_{j_i} =C$, where C is a already given value. For example, given $x={1,2,3,4}$ and $C=28$, then the permutation ${2,1,4,3}$ satisfies the above conditions.
If the above is not clear enough, essentially, the question is
Given a column vector $x$ of integer elements, and a constant $C$, find a permutation matrix $A$ such that $x^T A x=C$, where T denotes transpose.
$A$ is certainly not unique, so I am looking for a way to find all solutions.
I try to make $x$ into a square matrix by right multiply it by a row vector, but obviously this will certainly give a singular matrix without an inverse, so I make no progresses.
linear-algebra matrices discrete-mathematics permutations
edited Nov 15 at 4:44
gt6989b
32k22351
asked Nov 15 at 4:05
Ma Joad
799216
Suppose we have a sequence of integers ${x_i}$ and a permutation of this sequence changing its order ${x_{j_i}}$. We want to find a such a permutation making $sum_i x_ix_{j_i} =C$, where C is a already given value. For example, given $x={1,2,3,4}$ and $C=28$, then the permutation ${2,1,4,3}$ satisfies the above conditions.
If the above is not clear enough, essentially, the question is
Given a column vector $x$ of integer elements, and a constant $C$, find a permutation matrix $A$ such that $x^T A x=C$, where T denotes transpose.
$A$ is certainly not unique, so I am looking for a way to find all solutions.
I try to make $x$ into a square matrix by right multiply it by a row vector, but obviously this will certainly give a singular matrix without an inverse, so I make no progresses.
linear-algebra matrices discrete-mathematics permutations
linear-algebra matrices discrete-mathematics permutations
edited Nov 15 at 4:44
gt6989b
32k22351
asked Nov 15 at 4:05
Ma Joad
799216
edited Nov 15 at 4:44
gt6989b
32k22351
asked Nov 15 at 4:05
Ma Joad
799216
edited Nov 15 at 4:44
gt6989b
32k22351
edited Nov 15 at 4:44
gt6989b
32k22351
edited Nov 15 at 4:44
gt6989b
32k22351
32k22351
asked Nov 15 at 4:05
Ma Joad
799216
asked Nov 15 at 4:05
Ma Joad
799216
asked Nov 15 at 4:05
Ma Joad
799216
799216
add a comment |
add a comment |
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