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Royal Society of Chemistry eBooks

Matrices

Published Sep 12, 2025• 0 citations
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Abstract

Heavy use of matrices is made in the discussions of density operators and rotation operators. Thus the general properties of matrices are presented and, since they represent a non-Abelian group, some of their peculiarities are included. Their addition, subtraction and various kinds of multiplications are discussed. Determinants are developed from matrices and used in the solution of simultaneous equations. Various matrix transformations are discussed with a view towards their later use in density matrix calculations and finally, the exponentials of matrices are discussed.

DOI
10.1039/9781837678785-00010

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Royal Society of Chemistry eBooks•Sep 12, 2025•0 citations
Heavy use of matrices is made in the discussions of density operators and rotation operators. Thus the general properties of matrices are presented and, since they represent a non-Abelian group, some of their peculiarities are included. Their addition, subtraction and various kinds of multiplications are discussed. Determinants are developed from matrices and used in the solution of simultaneous equations. Various matrix transformations are discussed with a view towards their later use in density matrix calculations and finally, the exponentials of matrices are discussed.
Heavy use of matrices is made in the discussions of density operators and rotation operators. Thus, the general properties of matrices are presented and, since they represent a non-Abelian group, some of their peculiarities are included. Their addition, subtraction and various kinds of multiplications are discussed. Determinants are developed from matrices and used in the solution of simultaneous equations. Various matrix transformations are discussed with a view towards their later use in density matrix calculations and finally, the exponentials of matrices are discussed.
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Vectors are reviewed since they are extensively used in electric and magnetic field calculations and in quantum mechanics. A general definition of vectors is given followed by the more familiar Euclidean definition. The algebra of vectors is discussed including their addition, subtraction, multiplication and group characteristics. The matrix and Dirac notation for vectors is introduced and used to present vector transformations via operator matrices. A short description of Hilbert space and its fundamental definition is given. The merging of vectors spaces is discussed in preparation for the development of multiple-spin density matrices.
This chapter illustrates how one can obtain a matrix representation of a state vector, an operator, and an eigenvalue equation in a basis of orthonormal states. It helps to learn that matrix representations are very useful in calculations of the properties of operators, because algebra of matrices is very simple and completely developed. Readers familiar with the algebra of matrices agree that mathematical operations on matrices such as multiplication by a scalar, addition, subtraction, multiplication and diagonalization are simple and easy to perform. In fact, the application of the algebra of matrices to quantum physics results from the direct relations between operators and matrices that any mathematical relation that holds between operators also holds between the corresponding matrices. The eigenvalues and eigenvectors of the matrix representing a given operator  in the basis of orthonormal states are the same as the eigenvalues and eigenvectors of Â.
The solution of sets of linear simultaneous equations can be accomplished using matrices and we show how this is done. In quantum computation, matrices ‘operate’ or act on vectors to move a system from one quantum state to another as it evolves with time. The solution of sets of linear simultaneous equations can be accomplished using matrices and we show how this is done. Matrices can also be used to perform transformations of points in two- and three-dimensional space through rotations and reflections. Self-adjoint matrices play an important role in quantum computation because, as we shall see, they have real eigenvalues which correspond to the measured values of quantum observables. The procedure for calculating the inverse of a general n×n matrix necessitates the introduction of some new terminology. The rows of a unitary matrix are pairwise orthogonal so that the inner product of two distinct rows will be zero.
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