Abstract
Full Text
Physics
N. K. Morozova, V. P. Morozov
On the Reduction to Cell-Diagonal Form of Matrices of Operators Commuting with a Group of Symmetry Operators
(Presented by Academician I. V. Obreimov, 31 X 1964)
Let \(Q\) be a unitary space of dimension \(n\), whose elements are vectors
\[ \mathbf{q}=q_{1}\mathbf{e}_{1}+q_{2}\mathbf{e}_{2}+\cdots+q_{n}\mathbf{e}_{n} =\sum_{i=1}^{n}q_i\mathbf{e}_i; \tag{1} \]
here \(q_i\) are the natural coordinates defined in the sense of (1).
Let, further, \(G\) be a group of unitary symmetry operators \(\hat S^{(1)}, \hat S^{(2)}, \ldots \hat S^{(l)}\) (see \((2)\)), satisfying the condition \(\hat S^{*}=\hat S^{-1}\) (\(\hat S^{*}\) is the operator adjoint with respect to \(\hat S\)) and transforming each vector \(\mathbf{q}\) of the space \(Q\) into some vector \(\mathbf{q}'\) of the same space
\[ \mathbf{q}'=\hat S\mathbf{q}. \tag{2} \]
Suppose that the space of natural coordinates \(Q\) decomposes into subspaces of equivalent coordinates \(^{(1)}\) \(Q_1, Q_2, \ldots Q_k\), whose dimensions are respectively \(r_1, r_2, \ldots r_k\) \((n=r_1+r_2+\cdots+r_k)\).
From the definition of equivalence of coordinates it follows that each \(Q_j\) is invariant with respect to all symmetry operators. Then, in the basis \(\mathbf{e}_1, \mathbf{e}_2, \ldots, \mathbf{e}_n\), the symmetry operators \(\hat S^{(i)}\) correspond to matrices \(S^{(i)}\) having a stepwise structure
\[ S^{(i)}= \left\| \begin{array}{c:c:c:c} S_{1}^{(i)} & 0 & \cdots & 0\\ \hdashline 0 & S_{2}^{(i)} & \cdots & 0\\ \hdashline \vdots & \vdots & & \vdots\\ \hdashline 0 & 0 & \cdots & S_{k}^{(i)} \end{array} \right\| \qquad (i=1,2,\ldots,l), \tag{3} \]
where on the diagonal stand square matrices of orders \(r_1, r_2, \ldots, r_k\), respectively.
Let us now consider a Hermitian operator \(\hat A\)* which, in the orthonormal basis of the displacement space \(\mathbf{e}_1,\mathbf{e}_2,\ldots,\mathbf{e}_n\), has the matrix \(A=\|a_{ik}\|\), satisfying the invariance condition
\[ S^{(i)-1}AS^{(i)}=A, \tag{4} \]
where \(i=1,2,\ldots,l\).
Among all possible matrices satisfying condition (4), below we consider matrices possessing the least symmetry, i.e. the greatest possible number of essentially distinct elements \(a_{ik}\).
\[ \text{* Here by }\hat A\text{ is meant one of the following operators: } \hat T\text{ — the kinetic-energy operator, } \hat V\text{ — the potential-energy operator, and also } \hat T^{-1}\text{ and }\hat V^{-1}. \]
In accordance with the decomposition of the space \(Q\) into subspaces of equivalent coordinates, the matrix \(\|a_{ik}\|\) can be represented in block form:
\[ A= \begin{Vmatrix} \begin{array}{c:c:c:c} A_1 & A_{12} & \ldots & A_{1k}\\ \hdashline A_{21} & A_2 & \ldots & A_{2k}\\ \hdashline \vdots & \vdots & & \vdots\\ \hdashline A_{k1} & A_{k2} & \ldots & A_k \end{array} \end{Vmatrix}, \qquad \begin{array}{l} \}r_1\\[1.6em] \}r_2\\[1.6em] \vdots\\[1.6em] \}r_k \end{array} \tag{5} \]
where, from the self-adjointness of the operator \(\hat A\), it follows that
\[ A_{ik}^{*}=A_{ki}. \]
We shall now formulate the main proposition of the present work: if in each of the subspaces \(Q_j\) \((j=1,2,\ldots,k)\) one chooses, as a basis, the eigenvectors of the corresponding matrices \(A_j\), and then forms a basis of the space \(Q\) as the direct sum of the linear subspaces \(Q_j\) (see (3)), then in this new basis the matrix of the operator \(\hat A\), after a suitable permutation of rows and columns, will have block-diagonal form.
A justification of this proposition can be given as follows. Performing operations on the block matrices (3) and (5), we obtain, on the basis of (4),
\[ S_j^{(i)-1}A_jS_j=A_j \qquad (i=1,2,\ldots,l;\ j=1,2,\ldots,k), \]
whence
\[ A_jS_j^{(i)}=S_j^{(i)}A_j. \tag{7} \]
Let us note that the collection \(S_j^{(1)}, S_j^{(2)}, \ldots, S_j^{(l)}\) forms a representation \(G_j\) of the point symmetry group in the subspace of equivalent coordinates \(Q_j\).
We now construct the fundamental matrix (4) \(C_j\), whose columns are the eigenvectors of the operator \(\hat A_j\). Owing to the Hermitian nature of \(\hat A_j\), there always exists an orthonormal system of eigenvectors \((^{5})\). Then the similarity transformation brings the matrix \(A_j\) to diagonal form:
\[ C_j^{-1}A_jC_j=\|\lambda_s^{(j)}\delta_{st}\|=A_j'. \tag{8} \]
Suppose that among the \(\lambda_s\) there is an eigenvalue \(\lambda_p\) of multiplicity \(m\); to this eigenvalue there corresponds an \(m\)-dimensional eigensubspace \(L_p\). Then in the diagonal matrix \(A_j'\) one can distinguish a scalar submatrix \(A_p=\lambda_p E\) of order \(m\).
Thus the subspace of equivalent coordinates \(Q_j\) decomposes into the direct sum of the linear subspaces \(L_p\), whose dimensions are equal to the multiplicities of the corresponding eigenvalues \(\lambda_p\).
From the commutativity of \(A_j\) with all \(S_j^{(i)}\) it follows from (6) that the similarity transformation (8), performed on \(S_j^{(i)}\), leads to the decomposition of the representation \(G_j\)
\[ G_j=\sum_p \Gamma_p^{(j)}, \tag{9} \]
where \(\Gamma_p\) are the representations according to which the subspaces \(L_p\) transform, since the eigensubspaces of the operator \(\hat A_j\) are invariant subspaces of each \(S_j^{(i)}\). This follows from the fact that \(\hat A_j\) has a common complete system of eigenvectors with each of the \(S_j^{(i)}\) (4). For different \(i\) these will, generally speaking, be different systems.
Thus, the dimension of the representation \(\Gamma_p\) is equal to the multiplicity of the corresponding eigenvalue \(\lambda_p\).
Let us now show that the representations \(\Gamma_p\) are irreducible. This is obvious for the case of a one-dimensional eigensubspace \(L_p\) \((m=1)\), which will be an irreducible subspace in the sense of (7). Let us note in passing that the subspace \(Q_j'\) itself will be irreducible if its dimension is equal to one. Let us also note that if, on the right-hand side of (8), all eigenvalues are simple, then all \(\Gamma_p\) will be one-dimensional and irreducible and, consequently, the group \(G_j\) is abelian.
Consider the case when \(G_j\) is nonabelian and \(A_j'\) has at least one eigenvalue \(\lambda_p\) whose multiplicity is \(m \ne 1\). Suppose that the corresponding representation \(\Gamma_p\) is reducible. Then, in accordance with (6), there must exist a nonscalar submatrix \(B_p\) of order \(m\), commuting with \(\Gamma_p\). Now construct a matrix \(B_j'\) which differs from \(A_j'\) only in that, in place of the scalar matrix \(A_p'\), the matrix \(B_p\) stands. It is obvious that, by construction, \(B_j'\) commutes with all \(S_j^{(i)'}\)—the matrices of the reduced representation \(G_j'\). If, further, we carry out the transformation inverse to the transformation (8) on the matrix \(B_j'\), we obtain
\[ B_j=C_jB_j'C_j^{-1}, \tag{10} \]
where \(B_j\) must satisfy a condition analogous to (6), i.e.
\[ S_j^{(i)-1}B_jS_j^{(i)}=B_j. \tag{11} \]
From comparison of (11) and (6) it is clear that \(B_j\) must satisfy the condition of invariance with respect to the same group of matrices \(S_j^{(i)}\) as \(A_j\). But the matrix \(B_j\) must possess less symmetry than \(A_j\), in the sense that the number of essentially distinct elements of the matrix \(B_j\) must be greater than that of \(A_j\) (otherwise \(B_j\) would be reducible to diagonal form,* on the diagonal of which there would stand no more distinct eigenvalues than for \(A_j'\)). Thus we arrive at a contradiction with the choice of the matrix \(A_j\) as a matrix of least symmetry satisfying relation (6).
Thus, all eigensubspaces \(L_p\) of the operator \(\hat A_j\) are irreducible subspaces transforming according to irreducible representations \(\Gamma_p\).
Let us now construct the matrix \(C\).**
\[ C= \left\| \begin{array}{cccc} C_1 & 0 & \cdots & 0\\ 0 & C_2 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & C_k \end{array} \right\| \begin{array}{l} \}r_1\\ \}r_2\\ \\ \}r_k \end{array} \quad \begin{array}{cccc} \underbrace{\phantom{C_1}}_{r_1} & \underbrace{\phantom{C_2}}_{r_2} & \cdots & \underbrace{\phantom{C_k}}_{r_k} \end{array}, \tag{12} \]
in which each submatrix \(C_j\) \((j=1,2,\ldots,k)\) is fundamental for the submatrix \(A_j\) (4). The columns of the matrix (12) constitute the coordinates of the vectors \(\mathbf e_1', \mathbf e_2', \ldots, \mathbf e_n'\) (in the basis \(\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n\)), which, as is clear from the preceding discussion, form a basis of the completely reduced representation \(G\).
If we then perform the similarity transformation
\[ C^{-1}AC=A', \tag{13} \]
* From the commutativity \(A_pB_p=B_pA_p\) (\(A_p\) is scalar) it follows that \(A_j'B_j'=B_j'A_j'\) (by construction) and \(A_jB_j=B_jA_j\). Hence \(A_j\) and \(B_j\) have a complete common system of eigenvectors. Consequently, the transformation \(C_j'\) (not coinciding, generally speaking, with \(C_j\)) which brings \(B_j\) to diagonal form diagonalizes \(A_j\) to \(A_j'\).
** In forming the matrix \(C\) one must take into account the “orientation effect” (8).
then the matrix \(A'\), after a suitable permutation of rows and columns, is brought to block-diagonal form. The proof that the matrix of the operator \(\hat A\) in the basis in which \(G\) decomposes completely into irreducible representations is transformed to block-diagonal form can be carried out analogously to the way this was done in (2) for the vibration operator \(\hat W\).
In this case the coordinates of the vector \(\mathbf q\) (see (1)) are transformed in accordance with the equality
\[ \xi = C^{-1}q, \tag{14} \]
where \(q\) is the column matrix of the coordinates of the vector \(\mathbf q\) in the basis \(e_1, e_2, \ldots, e_n\), and \(\xi\) is the column matrix of the coordinates of the same vector in the basis \(e'_1, e'_2, \ldots, e'_n\); \(\xi_i\) are the symmetry coordinates \((^{1,2,8})\).
On the basis of what has been set forth, one may give the following definition: symmetry coordinates are the coordinates of the vector \(\mathbf q\) in a basis composed of eigenvectors of submatrices of equivalent coordinates.
Thus, the matrix \(C\) can be obtained directly from the properties of the matrix \(A\), without invoking the theory of representations of symmetry groups. However, on the basis of the results obtained it is easy to find the irreducible representations of the symmetry group; for this it suffices to write the matrices of the operators \(S^{(i)}\) in the new basis.
In conclusion, let us note that the vibration operator \(\hat W = \hat T^{-1}\hat V\), which is not Hermitian, is reduced, by symmetry, by the same transformation as \(\hat A\) (see (1)).
Dnepropetrovsk Chemical-Technological Institute
named after F. E. Dzerzhinsky
Received
31 X 1964
CITED LITERATURE
- M. V. Vol'kenshtein, M. A. El'yashevich, B. I. Stepanov, Vibrations of Molecules, Moscow–Leningrad, 1948.
- L. S. Mayants, Theory and Calculation of Molecular Vibrations, Publishing House of the Academy of Sciences of the USSR, 1960.
- A. I. Mal'tsev, Foundations of Linear Algebra, Moscow–Leningrad, 1948.
- F. R. Gantmacher, The Theory of Matrices, Moscow, 1954.
- I. M. Gel'fand, Lectures on Linear Algebra, Moscow–Leningrad, 1948.
- E. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Foreign Literature Publishing House, 1961.
- T. Ya. Lyubarskii, Group Theory and Its Application in Physics, Moscow, 1958.
- E. B. Wilson, J. C. Decius, P. C. Cross, Molecular Vibrations, Foreign Literature Publishing House, 1960.