Unity matrix

It is expected that for a certain choice of paiameters (that define the x matrix) the adiabatic-to-diabatic transformation matrix becomes identical to the corresponding Wigner rotation matrix. To see the connection, we substitute Eq. (51) in Eq. (28) and assume A( o) to be the unity matrix. 

By a simple linear transformation, the core matrix becomes a unity matrix (Table 2). 

The next step is to transform the core matrix into a unity matrix. Hence, the order of the physical variables in the core matrix should be such that a minimum amount of linear transformations is required. Adding the M row to the L and T rows eliminates the non-zero term below the diagonal in the core matrix, i.e.,... 

The mathematical description of hybridization is based on employing the algebraic group structure of the hybrids manifold. Because of it, any small variation of HOs in a vicinity of a given set of HOs given by a 4 x 4 orthogonal matrix h can be expressed using another 50(4) matrix H close to the unity matrix ... 

Obviously the zero values of the angular variables define the unity matrix of the 50(4) group ea is the complete antisymmetric - Levi-Civita - tensor. The commutation properties of the infinitesimal operators defined in this manner are... 

The number of variables can be easily reduced in the vicinity of the unity matrix, which is of real interest to us, without the loss of the rational character of dependence on variables. The 50(4) matrix H close to the unity matrix I has an expansion ... 

The unity matrix obviously corresponds to a pair of quaternions with ... 

Since the core matrix has to be transformed into a unity matrix (zero-free main diagonal, otherwise zeros) the fillers should be arranged in such a way as to facilitate a minimum of linear transformations. 

Now, only one further linear transformation of the rows is necessary to transform the core matrix into a unity matrix, see the matrix below. 

When generating dimensionless numbers, each element of the residual matrix forms the numerator of a fraction while its denominator consists of the fillers from the unity matrix with the exponents indicated in the residual matrix, see eq. (3.2). 

A conveniently constructed dimensional matrix (for dimensions see Table 3) contains a core matrix which is transformed in three steps into the unity matrix. It just so happens that in the last linear transformation the 1. row and the p-columns eliminate. This represents a reduction of the rank, r, of the matrix. 

The following - conveniently drawn up - dimensional matrix leads to, after only two linear transformations, the unity matrix (rank r = 3) and the residual matrix ... 

If M is a diagonal matrix (experimental errors, but no correlations), M-1 is called the weight matrix, and the solution (Eqs. 21) is that for the weighted, uncorrelated least-squares problem. If M is introduced in Eqs. 21 as the unity matrix, Eqs. 21 solve the equally (unity-) weighted, uncorrelated problem. In this special case, the covariance matrices Eqs. 21b and 21e) depend on the measurements only through the common factor a2 (Eq. 21c). 

Transformation of the dimensional matrix (Table 3) into a unity matrix is straightforward. To transform-3 in L-row/p-column into zero, one linear transformation is required. The subsequent multiplication of the T-row by —1 transfers the —1 of the n-column to +1. 

A diagonal matrix in which all diagonal elements are equal to one, a = I for all i, is called i unity matrix or an identity matrix. It is denoted by I. Sometimes the dimension is specified by an index. 

Here, is the unity matrix of Hilbert space. In the case of the tunnel Hamiltonian it is simply... 

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