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Royal Society of Chemistry eBooks

Matrices

Published Dec 13, 2016• 0 citations
Publisher

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/bk9781782627975-00012

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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 notations for vectors are 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 vector spaces is discussed in preparation for the development of multiple-spin density matrices.
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