Unitary matrix definition

The D matrix is by definition a unitary matrix (it is product of two unitary mati ices) and since the adiabatic eigenvalues are uniquely defined in CS, we have, u(0) = m(P). Then, Eq. (57) can be written as... 

The B matrix is, by definition, a unitary matrix (it is a product of two unitary matrices) and at this stage except for being dependent on F and, eventually, on So, it is rather arbitrary. In what follows, we shall derive some features of B. 

As V is a unitary matrix, Y = VTX is just an equivalent set of Cartesian coordinates, and = UTZ is just an equivalent set of internal coordinates, simply linear combinations of the Zn. The i, , N-6, change independently, in proportion to changes in linear combinations of the Cartesian coordinates. So, locally, we have defined 3N — 6 independent internal coordinates. Every different configuration of the molecule, X, will have a different B matrix, and hence a different definition of local internal coordinates, defined automatically. 

Exercise 11.2 (Used in Section 11.2) Suppose M is a Hermitian-symmetric, finite-dimensional matrix (as defined in Exercise 3.25). Show that there exists a real diagonal matrix D and a unitary matrix B (see Definition 3.5) such that... 

The other theorem states that the matrix X formed by using the eigenvectors of a Hermitian matrix as its columns is unitary (for the definition of a unitary matrix, see Appendix A.4-l(ff)). The proof of these two theorems is given in Appendix A.4-3. 

It follows from the definition of a unitary matrix that detU == 1. As for orthogonal matrices, an arbitrary matrix having the property detU = 1 is unitary only if the requirements of equation (4.29) are satisfied. 

Various methods of constructing a unitary matrix V with localizing properties have been proposed [for a review, see Ref.45>]. For the present analysis we have adopted Boys method4 ), as being the simplest intrinsic method (for a definition of intrinsic versus external methods, see Ruedenberg47)]. 

Here Fdimer and Sdimer have the same definitions as before, the unitary matrix U is formed from the solutions Uj of the generalized eigenvalue equation... 

Two elements belonging to the same column of a unitary matrix U, are Uiy and Ut). By definition... 

A unitary matrix has several interesting properties, which can easily be checked from the general definition ... 

We now wish to diagonalize D and U in order to find linear combinations of the bare quarks which have definite mass. Since D (and U) is an arbitrary matrix it cannot be diagonalized by one single unitary matrix as would be the case for a hermitian matrix. But it is possible to find pairs of unitary matrices U, U and C/l,such that... 

For the real unitary matrix representations we have been considering, the matrix of the inverse transformation is obtained from the original matrix by simply interchanging rows and columns. By definition, therefore... 

Definition A matrix U is said to be unitary if t/ = 17. A real unitary matrix Q is... 

For a positive definite matrix the use of a unitary factor should be emphatically ruled out. For triangular factorization, if... 

The 5-matrix is unitary and symmetric, while the T-matrix is symmetric. This particular definition of the T-matrix reduces for scattering by a central potential to the phase-shift factor in the scattering amplitude,... 

Some types of matrices that appear frequently are Hermitian matrices, for which A = A anti-Hermitian matrices, for which A = — A and unitary matrices, for which = U The MCSCF methods diseussed in most detail in this review will involve only operations of real matrices. In this case these matrix types reduce to symmetric matrices, for which A = A , antisymmetric matrices, for which A = — A, and orthogonal matrices, for which U = U. A particular type of orthogonal matrix is called a rotation matrix and satisfies the relation Det(R)= -I-1, where Det(R) is the usual definition of a determi-... 

A matrix is unitary if its rows and columns are orthonormal. In this definition the scalar product of two rows (or two columns) is obtained by adding pairwise products of the corresponding elements, AijAtj, one of which is taken to be complex conjugate ... 

A matrix product of the form A" HA is called a similarity transformation on H. If A is orthogonal, then AHA is a special kind of similarity transformation, called an orthogonal transformation. If A is unitary, then A HA is a unitary transformation on H. There is a physical interpretation for a similarity transformation, which will be discussed in a later chapter. For the present, we are concerned only with the mathematical definition of such a transformation. The important feature is that the eigenvalues, or latent roots, of H are preserved in such a transformation (see Problem 9-5). 

The first stages of the procedure are precisely the same as for the exact Foldy-Wouthuysen transformation, which we will follow here. The definitions in the paper by Barysz et al. differ by a sign in the lower row of the transformation matrix, but the results obviously do not depend on this sign. We write the transformation matrix in similar form to the formal unitary Foldy-Wouthuysen transformation ... 

The definition and properties of such operators are presented here.in general form. In the text they have been used mainly for unitary representations, in which case the matrix elements in the contragredient representation (p. 538) ate D R j =... 

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