Unimodular transformation

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... 

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 ... 

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 ... 

A transformation A (B.11.20) in (B.11.21) is called unimodular (volume-preserving) when detA = 1. In particular any orthonormal transformation, say A = Q is unimodular because QQ = If., thus (detQ) =(detQ)(detQ ) = 1 thus detQ = 1. As a special case, imagine rotation in 3-dimensional space, preserving clearly the volume. 

A further reduction occurs when we use only real unimodular matrices we then obtain the real orthogonal group R(m), in which U = U, and the distinction between co- and contravariant transformations disappears. 

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