Matrix transformation elementary similarity

It is often desirable to transform a matrix to a different form which is more amenable to solution. There are several such transformations that convert matrices without significantly changing their properties. We will divide these transformations into two categories elementary transformations and similarity transformations. 

The complete elementary similarity transformation that converts matrix A to the upper Hessenberg matrix H is shown by... 

The elementary similarity transformation to produce an upper Hessenberg matrix in formula form is as follows ... 

The M-dimensional adiabatic-to-diahatic transformation matrix will be written as a product of elementary rotation matrices similar to that given in Eq. (80) [9] ... 

Our theorem permits the following inference. The statistical matrix of every pure case in quantum mechanics is equivalent to an elementary matrix and can be transformed into it by a similarity transformation. Because p is hermitian, the transforming matrix is unitary. A mixture can, therefore, always be written in the diagonal form Eq. (7-92). 

Let us now show that the Jordan canonical form is similar to a matrix in the canonical form N [Eq. (183)]. The elementary Jordan matrices are transformed into the required form by... 

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