Vector spaces, diagonal, algebra

In the course of study, students should master material that is both simple and complex. Much of this involves familiarity with the set of mathematical tools repeatedly used throughout this book. The Appendices provide ample reference to such a toolbox. These include matrix algebra, determinants, vector spaces, vector orthogonalization, secular equations, matrix diagonalization, point group theory, delta functions, finding conditional extrema (Lagrange multipliers, penalty function methods), Slater-Condon rules, as well as secondary quantization. 

Two important complex numbers associated to any particular complex linear operator T (on a finite-dimensional complex vector space) are the trace and the determinant. These have algebraic definitions in terms of the entries of the matrix of T in any basis however, the values calculated will be the same no matter which basis one chooses to calculate them in. We define the trace of a square matrix A to be the sum of its diagonal entries ... 

In quantum mechanics we often encounter associative algebras of operators and matrices which are noncommutative. For example, the set of all n x n matrices over the real or complex number fields is an n2-dimensional vector space which is also an associative, noncommutative algebra whose multiplication is just the usual matrix multiplication. Also, the subset of all diagonal n x n matrices is a commutative algebra. 

The operators act on the space of smooth vector functions, and the Lie algebra Afn i is realized by square zero-trace matrices. The matrices a and b are diagonal with distrinct diagonal elements. According to the finite-zoned integration theory (see [77]), the commutativity equations [Lai Aa] = 0 are integrated by means of the theta-functions of the Riemann surface of the algebraic curve Q W A) = det(lV — X - Aa) = 0. 

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