Find a permutation making the sum of products to equal a given value
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
31.9k22351
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
31.9k22351
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
31.9k22351
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
31.9k22351
asked Nov 15 at 4:05
Ma Joad
799216
edited Nov 15 at 4:44
gt6989b
31.9k22351
asked Nov 15 at 4:05
Ma Joad
799216
edited Nov 15 at 4:44
gt6989b
31.9k22351
edited Nov 15 at 4:44
gt6989b
31.9k22351
edited Nov 15 at 4:44
gt6989b
31.9k22351
31.9k22351
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 |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2999197%2ffind-a-permutation-making-the-sum-of-products-to-equal-a-given-value%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
