Special unitary matrices

The exponential parametrization of a unitary matrix in (3.1.9) is a general one, applicable under all circumstances. We shall now consider more special forms of unitary matrices. We begin by writing the anti-Hermitian matrix X in the form 

It can be demonstrated that any unitary matrix can be written as a unitary diagonal matrix times a special unitary matrix [1] 

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Exercise 4.3 Show that the set ofly. diagonal special unitary matrices is a group and that it is isomorphic to the group T x T. (See Exercise 4.1 for the definition of the Cartesian product of groups.)... 

The unitary diagonal matrix exp [i d ] induces phase shifts of the orbitals. These phase shifts are redundant in our future derivations where only special unitary matrices (Eq. (4.15)) need to be considered. In time-independent theory it is practise to use real orbitals. All rotations of real orbitals into real orbitals can be written... 

SU(n) Group Algebra. Unitary transformations, U( ), leave the modulus squared of a complex wavefunction invariant. The elements of a U( ) group are represented by n x n unitary matrices with a determinant equal to 1. Special unitary matrices are elements of unitary matrices that leave the determinant equal to +1. There are n2 — 1 independent parameters. SU( ) is a subgroup of U(n) for which the determinant equals +1. 

An elegant description of rotation in spherical mode is provided in terms of a special unitary matrix of order 2, known as SU(2) in Lie-group space (T2.8.2). The matrices that form a basis for the algebra of SU(2) are those already introduced to represent quaternions. The important result is that the group space SU(2) is compact compared to a noncompact group R that characterizes cylindrical rotation about an axis of infinite extent. If an object... 

The derivation of the D matrix for a given contour is based on first deriving the adiabatic-to-diabatic transformation matrix, A, as a function of s and then obtaining its value at the end of the arbitrary closed contours (when s becomes io). Since A is a real unitary matrix it can be expressed in terms of cosine and sine functions of given angles. First, we shall consider briefly the two special cases with M = 2 and 3. 

An orthogonal matrix is one whose inverse is equal to its transpose A, =A. A unitary matrix is one whose inverse is equal to its Hermitian conjugate A 1 = A. A real orthogonal matrix is a special case of a unitary matrix. 

Finally, using the eigenvalues there are some further subdivision possible If the product of eigenvalues of a unitary matrix or operator is equal to +1, it is called a special unitary (SU) matrix or operator. Similar for real orthogonal matrices, where the only possible choice is +1 or -1 the former case is called special orthogonal (SO) matrices. For a matrix, this product equals the determinant of the matrix. For both matrices and operators, the sum of eigenvalues is called the trace of the matrix or operator. This equals the sum of the diagonal elements of a matrix representation. 

Special orthogonal matrices such as Householder matrices H = Im — 2vv for a unit column vector v G Cm with v = v v = 1 can be used repeatedly to zero out the lower triangle of a matrix Amn much like the row reduction process that finds a REF of A in subsection (B). The result of this elimination process is the QR factorization of Am,n as A = QR for an upper triangular matrix Rm,n and a unitary matrix Qm,m that is the product of n — 1 Householder elimination matrices Hi. 

Unfortunately, in the presence of F(r), the unitary matrix giving the exact decoupling is not found in a closed form. A number of different approximations to the exact FW transformation have been suggested and analyzed in the literature. - With the special choice of approximations to the exact decoupling, the effective two-component ZORA Hamiltonian in the presence of electromagnetic fields is... 

The last results we require will simply be stated and verified by examples. Every unitary matrix and every Hermitian matrix can be diagonalized by a similarity transformation with a unitary matrix. The unitary matrix is rather special in the. sense that its columns are not only vectors but they are orthogonal vectors. 

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... 

Another special class of matrix, the unitary matrix, A = A+, has characteristic values of absolute value unity, X X = 1. Consider its secular equation, written in matrix form... 

A special unitary transformation exp(—S) is obtained by omitting the diagonal elements of this expression see Section 3.1.3. Moreover, an orthogonal transformation of the configuration states is obtained by employing in exp(—S) a real antisymmetric matrix S. 

The implicit-midpoint (IM) scheme differs from IE above in that it is symmetric and symplectic. It is also special in the sense that the transformation matrix for the model linear problem is unitary, partitioning kinetic and potential-energy components identically. Like IE, IM is also A-stable. IM is (herefore a more reasonable candidate for integration of conservative systems, and several researchers have explored such applications [58, 59, 60, 61]. 

The equations may be simplified by choosing a unitary transformation (Chapter 13) which makes the matrix of Lagrange multipliers diagonal, i.e. Ay 0 and A This special set of molecular orbitals (f> ) are called canonical MOs, and they transform eq. (3.40) mto a set of pseudo-eigenvalue equations. 

Several years ago Baer proposed the use of a matrix A, that transforms the adiabatic electronic set to a diabatic one [72], (For a special twofold set this was discussed in [286,287].) Computations performed with the diabatic set are much simpler than those with the adiabatic set. Subject to certain conditions, A is the solution of a set of first order partial differential equations. A is unitary and has the form of a path-ordered phase factor, in which the phase can be formally written as... 

Note that the matrix in this formula is unitary. This completes the proof in the special case where 5 fixed the point [e ] and every element of [ ]. 

In this section we shall present a simple unified approach to the matrix representation theory of the so(2,1) and so(3) Lie algebras (Barut and Fronsdal, 1965 Barut, 1971 Wybourne, 1974 Barut and Raczka, 1977). These Lie algebras have a similar structure and the general representation theory for both are closely related. However, when we specialize to the unitary representations by requiring that the generators be Hermitian, so(2, 1) and... 

An important special case of this analysis occurs when the dimension of the B matrix is unity. Without loss of generality the matrix M may be considered to be diagonal. (It may always be brought to diagonal form with a suitable block-diagonal unitary transformation.) In this case L(A) is a single-valued function of the form... 

Being an angular momentum, the spin should be associated with the symmetry of the rotation group. Thus the rotation group can be described by Special (their determinant is 1) Unitary 2x2 matrices and so this group is called SU(2). The determinant of matrix A is... 

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). 

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