The name is absent



14

where ɑʃ = (аг1,..., α⅛) is the г-th row of A for i N, and b = (Z>1,..., δra+1)τ.

A linear transformation U is called a unimodular transformation if U is bijective
on
Zn. U is unimodular if and only if the entries of U are integral and the deter-
minant of
U is equal to 1 or —1. Let A be the product AU. If U is unimodular,
it is readily seen that given an integral point
x and у = Ux, Ay ≤ b if and only if
Ax ≤ b.

One may wonder how to get a unimodular matrix. Clearly, the identity In is
unimodular. In order to obtain a non-trivial unimodular matrix, however, we need
to study some other examples and their effect when they postmultiply a matrix
A
and premultiply a vector x:

(г) Interchange: U is equal to In except that the Avth column of U is e(Z) and the
Z-th column of
U is e(k^). This transformation switches columns к and I of A
and switches the Avth and Z-th components of x.

(U) Reversal of sign: Z7 is equal to In except that the Avth column of Z7 is equal
to
-e(k^). U changes the sign of the entries of the Ат-th column of A and of
the Ат-th component of
x.

(ггг) Addition: Z7 is equal to In except that the (A:, Z)-th entry of Z7, к ψ Z, is equal to
one. This transformation replaces the Z-th column of
A by the sum of columns
к and Z and replaces the Ат-th component of x by the sum of components к
and I.

The following result can be found in Newman [9].

Theorem 4.1 Every unimodular matrix can be expressed as a finite product of
unimodular matrices of type
(г), (гг) and (ггг).

The next basic result says that every simplex can be brought into the standard form
by a unimodular transformation.



More intriguing information

1. Female Empowerment: Impact of a Commitment Savings Product in the Philippines
2. Meat Slaughter and Processing Plants’ Traceability Levels Evidence From Iowa
3. The name is absent
4. Input-Output Analysis, Linear Programming and Modified Multipliers
5. Campanile Orchestra
6. Improvement of Access to Data Sets from the Official Statistics
7. GROWTH, UNEMPLOYMENT AND THE WAGE SETTING PROCESS.
8. Internationalization of Universities as Internationalization of Bildung
9. Spectral density bandwith choice and prewightening in the estimation of heteroskadasticity and autocorrelation consistent covariance matrices in panel data models
10. The name is absent