Unitary matrix exponential form

A usual representation of a unitary matrix is the exponential form ... 

The Elementary Jacobi Rotations can be taken as an alternative form for the unitary matrix transformation of the multiconfiguration energy. The exponential and the EJR forms of the unitary matrix can be related through the expressions,already discussed in section 2.5, because the EJR are the minimal parts in which the exponential... 

In the literature, many forms of the unitary matrix in Eq. (11.27) can be found and are not easy to be recognized as the same — often also because the notation is varied and authors chose not to stick to historically older conventions. To demonstrate how these different forms can be interconverted we may consider some of them explicitly in the following. An often employed version of Uq, identical to the one above, is the exponential form. 

The elements of the unitary group are continuous functions of a certain number of parameters. The number of independent parameters for U(n) is n. This can be seen by writing matrix U in the exponential form ... 

Since iV8V is anti-Hermitian, we have shown that any unitary matrix can be written in the exponential form (3.1.9). 

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... 

Thus the EJR can be viewed as the simplest form of an exponential unitary transformation matrix, expressed in a real space. 

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