Determinant unimodular

It is known as the unitary unimodular group, or the special unitary group denoted by SU(2). Because of the extra condition on the determinant, SU(2) is a three-parameter group. 

Matrices which represent proper rotations are unimodular, that is they have determinant+ 1 and are unitary (orthogonal, if the space is real, as is ft3). Consider the set of all 2 x 2 unitary matrices with determinant +1. With binary composition chosen to... 

Rotations in a vector space of three orbitals are described by the group SO(3) of orthogonal 3x3 matrices with determinant +1. To embed the octahedral rotation group in this covering group one needs a matrix representation of O which also consists of orthogonal and unimodular 3x3 matrices. Such a matrix representation is sometimes called the fundamental vector representation of the point group. In the case of O the fundamental vector representation is Ti and not T2. Indeed the 7] matrices are unimodular, i.e. have determinant +1, while the determinants of the T2 matrices are equal to the characters of the one-dimensional representation A2. 

The unimodular condition requires only a normalizing factor on the matrix. Each term must be divided by the square root of the determinant,... 

A square matrix is unimodular if its determinant is 1. unit circle... 

A noteworthy property of matrices found in Eqs. 1.21, 1.22 and 1.24 to 1.26 is their unimodularity - the determinant of every matrix is equal to 1 or -1 for the rotation and inversion (or roto-inversion) operations, respectively, which is shown for the rotation around Z in Eq. 1.27. 

It is a well known theorem that that the value of a determinant is unchanged by unimodular linear transformations among its rows or columns and, of course, the quotient eqn ( 2.1) is unchanged by any non-singular linear transformation among the Xi- In particular, we may choose the Xt to be an orthonormal set and therefore take advantage of the enormous simplification in the expressions for the numerator and denominator of eqn ( 2.1) in this case ... 

It is both conceptually and computationally useful to cast the equation into a form which can be interpreted more easily. This possibility hinges on a simple property of determinants the fact that the value of a determinant is invariant against a unimodular transformation amongst its rows (or columns). The molecular orbitals of which the single-determinant wavefunction is composed may be... 

The most common of these exact cases are optimization problems that can be modeled as singlecommodity network flows (see Chapter 99). Equivalently, these are the (ILP)s that can be written so that for each variable Xj, at most one constraint coefficient A j equals 1, at most one A j equals —1, and all other equal 0. Such ILP) s are totally unimodular in that any submatrix formed by the Ajj associated with a collection of rows i and a like-sized collection of variables y has determinant 0, 1 or — 1. This is enough to ensure optimal basic solutions to (/LP) (produced, for example, by the simplex algorithm for linear programming) are integer whenever right-hand-side coefficients are all integer. 

To establish the connection between the spinor and the vector, we now need to verify how transformations in the spinor are manifested as transformations in the vector. Consider a finite unitary transformation of the spinor. The transformation belongs to the unitary group, U(2), and, as we have seen, the determinant of this matrix is unimodular. We consider the special case, however, where the determinant is +1. Such matrices form the special unitary group, SU 2). The most general form of an SU(2) matrix involves two complex parameters, say a and b, subject to the condition that their squared norm, a + b, equals unity. These parameters are also known as the Cayley-Klein parameters. (Cf. Problem 2.1.) One has... 

The interesting aspect of these side constraints is how they impact the computational complexity of the winner determination problem. For example, introducing either of the following constraint classes will transform even a tractable problem (e.g. with a totally unimodular structure) into a hard problem ... 

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