Monge arrays, or
Monge matrices, are mathematical objects used primarily in
computer scienceComputer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe and transform...
. They are named for their discoverer, the French mathematician
Gaspard MongeGaspard Monge, Comte de Péluse , was the a French mathematician and inventor of descriptive geometry.-Biography:...
.
An
m-by-
n matrixIn mathematics, a matrix is a rectangular array of numbers, such asEntries of a matrix are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied...
is said to be a
Monge array if, for all such that
one obtains
So whenever we pick two rows and two columns of a Monge array (a 2x2 sub-matrix) and consider the four elements at the intersection points, the sum of the upper-left and lower right elements (across the
main diagonalIn linear algebra, the main diagonal of a matrix is the collection of cells where is equal to ....
) is less than or equal to the sum of the lower-left and upper-right elements (across the antidiagonal).
This matrix is a Monge array:
For example, take the intersection of rows 2 and 4 with columns 1 and 5.
The four elements are:
- 17 + 7 = 24
- 23 + 11 = 34
The sum of the upper-left and lower right elements is less than or equal to the sum of the lower-left and upper-right elements.
- The above definition is equivalent to the statement
- A matrix is a Monge array if and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
for all and .
- Any subarray produced by selecting certain rows and columns from an original Monge array will itself be a Monge array.
- Any linear combination
In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.- Definition:Suppose that K is a...
with non-negative coefficients of Monge arrays is itself a Monge array.
- One interesting property of Monge arrays is that if you mark with a circle the leftmost minimum of each row, you will discover that your circles march downward to the right; that is to say, if , then for all .
Discussion
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Monge arrays, or
Monge matrices, are mathematical objects used primarily in
computer scienceComputer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems. It is frequently described as the systematic study of algorithmic processes that create, describe and transform...
. They are named for their discoverer, the French mathematician
Gaspard MongeGaspard Monge, Comte de Péluse , was the a French mathematician and inventor of descriptive geometry.-Biography:...
.
An
m-by-
n matrixIn mathematics, a matrix is a rectangular array of numbers, such asEntries of a matrix are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible size can be multiplied...
is said to be a
Monge array if, for all such that
one obtains
So whenever we pick two rows and two columns of a Monge array (a 2x2 sub-matrix) and consider the four elements at the intersection points, the sum of the upper-left and lower right elements (across the
main diagonalIn linear algebra, the main diagonal of a matrix is the collection of cells where is equal to ....
) is less than or equal to the sum of the lower-left and upper-right elements (across the antidiagonal).
This matrix is a Monge array:
For example, take the intersection of rows 2 and 4 with columns 1 and 5.
The four elements are:
- 17 + 7 = 24
- 23 + 11 = 34
The sum of the upper-left and lower right elements is less than or equal to the sum of the lower-left and upper-right elements.
Properties
- The above definition is equivalent to the statement
- A matrix is a Monge array if and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
for all and .
- Any subarray produced by selecting certain rows and columns from an original Monge array will itself be a Monge array.
- Any linear combination
In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics.Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.- Definition:Suppose that K is a...
with non-negative coefficients of Monge arrays is itself a Monge array.
- One interesting property of Monge arrays is that if you mark with a circle the leftmost minimum of each row, you will discover that your circles march downward to the right; that is to say, if , then for all . Symmetrically, if you mark the uppermost minimum of each column, your circles will march rightwards and downwards. The row and column maxima march in the opposite direction: upwards to the right and downwards to the left.
- The notion of weak Monge arrays has been proposed; a weak Monge array is a square n-by-n matrix which satisfies the Monge property only for all .
Applications
- A square Monge matrix which is also symmetric about its main diagonal
In linear algebra, the main diagonal of a matrix is the collection of cells where is equal to ....
is called a Supnick matrixA Supnick matrix or Supnick array – named after Fred Supnick of the City College of New York, who introduced the notion in 1957 – is a Monge array which is also a symmetric matrix.- Mathematical definition :...
(after Fred Supnick); this kind of matrix has applications to the traveling salesman problem (namely, that the problem admits of easy solutions when the distance matrixIn mathematics, a distance matrix is a matrix containing the distances, taken pairwise, of a set of points. It is therefore a symmetric N×N matrix containing non-negative reals as elements, given N points in Euclidean space...
can be written as a Supnick matrix). Note that any linear combination of Supnick matrices is itself a Supnick matrix.