Matrix transformation similarity

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] ... 

The unitary transform does the same thing as a similarity transform, except that it operates in a complex space rather than a real space. Thinking in terms of an added imaginary dimension for each real dimension, the space of the unitary matrix is a 2m-dimensionaI space. The unitary transform is introduced here because atomic or molecular wave functions may be complex. 

We have found the principal axes from the equation of motion in an arbitrary coordinate system by means of a similarity transformation S KS (Chapter 2) on the coefficient matrix for the quadratic containing the mixed terms... 

Based on the above similarity transform, we ean now show that the traee of a matrix (i.e., the sum of its diagonal elements) is independent of the representation in whieh the matrix is formed, and, in partieular, the traee is equal to the sum of the eigenvalues of the matrix. The proof of this theorem proeeeds as follows ... 

We will see later that performing a similarity transform expresses a matrix in a... 

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). 

This is, therefore, the form taken by a similarity transformation of co-representation matrices of nonunitary groups, and the two sets of matrices D and B are considered to be equivalent. It is interesting to note that if one lets V = o>E be a multiple of the unit matrix B(i)(u) =... 

Suppose now that we could find a similarity transformation such that N AiN=Ai should be a diagonal matrix. It is then obvious that A2, A3, A and 5 being simple polynomials in Ai, should be diagonal, too, because... 

This emphasizes that the monodromy matrix is defined only within a similarity transformation... 

In addition one can always find a transformation leading to a symmetry adapted basis [4] e, so that T is brought to the block diagonal form T via the associated similarity transformation. The T matrix can be written as a direct sum... 

This transformation of the matrix A to a diagonal matrix is an example of a similarity transform. 

In order to fulfill compatibility condition (a), the local coordinate system of each parent molecule Mk can always be reoriented, resulting in a simple similarity transformation of the original fragment density matrix P (qS(Kk)) into a compatible fragment density matrix P (cp (K)),... 

A tool that we should be familiar with from introductory linear algebra is similarity transform, which allows us to transform a matrix into another one but which retains the same eigenvalues. If a state x and another x are related via a so-called similarity transformation, the state space representations constmcted with x and x are considered to be equivalent.1... 

We first do a demonstration of similarity transform. For a nonsingular matrix A with distinct eigenvalues, we can find a nonsingular (modal) matrix P such that the matrix A can be transformed into a diagonal made up of its eigenvalues. This is one useful technique in decoupling a set of differential equations. 

Hence, as is often stated, the determination of the normal coordinates is equivalent to the successful search for a matrix L that diagonalizes the product GF via a similarity transformation. This system of linear, simultaneous homogeneous equations can be written in the form... 

Thih result shows that the tnatrix (SRS 1) in the new basis corresponds to R (he original one. The relation between them is a similarity transformation (see Section 7.10). It is now necessary to demonstrate that the character of a matrix transformation is invariant under a similarity transformation. 

The obvious way to form a similarity between the Wigner rotation matrix and the adiabatic-to-diabatic transformation matrix defined in Eqs. (28) is to consider the (unbreakable) multidegeneracy case that is based, just like Wigner rotation matrix, on a single axis of rotation. For this sake, we consider the particular set of x matrices as defined in Eq. (51) and derive the relevant adiabatic-to-diabatic transformation matrices. In what follows, the degree of similarity between the two types of matrices will be presented for three special cases, namely, the two-state case which in Wigner s notation is the case, j = j, the tri-state case (i.e.,j = 1) and the tetra-state case (i.e., j = ). 

A similarity transformation is effected by multiplying a matrix by another matrix and its inverse to produce yet another matrix, according to... 

Now suppose that B is a diagonal matrix (all off-diagonal elements equal to zero) then the roots of its characteristic equation (eigenvalues) are identical with its diagonal elements. If A is not a diagonal matrix but is related to B by a similarity transformation, it follows that it has the same characteristic equation and roots as B. The problem of finding the eigenvalues... 

It can be shown that the new matrix after the similarity transform... 

Note that = det is the determinant of the 3N x 3N transformation matrix 8R /8g , which gives the Jacobian for the transformation from generalized to Cartesian coordinates. This follows from the fact that the right-hand side (RHS) of Eq. (2.16) for g p is a matrix product of this transformation matrix with its transpose, and that the determinant of a matrix product is a product of determinants. By similar reasoning, we find that... 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

In the case of the representation of C3 in Figure 6.2, it is obvious that the matrices are merely combinations of of simpler representations. But if the three matrices were subjected to a similarity transformation, they would no longer be in block form, and it would not be obvious that the representation is composite. Applying the reverse similarity transformation would put the matrices back into block form. If there exists a similarity transformation such that applying it to each matrix in a representation puts every matrix into congruent block form, the representation is said to be reducible. If no such similarity transformation exists, the representation is said to be irreducible. 

This representation is in block form, and is obviously reducible. Consider another coordinate system, rotated in the a — y plane by 45°. Verify that in this new coordinate system the formulas giving the effect of cr are a —y and y —s- —x. Find the matrix relating the two coordinate systems and verify that a similarity transformation applied to the matrices of this new representation produces the old representation. How does this demonstrate the reducibility of the new representation ... 

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