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Springer eBooks

Advanced Matrix Properties

James K. Freericks

Published Jan 1, 2026• 0 citations
Publisher

Abstract

As we move into more advanced properties of quantum mechanics, we need to extend our understanding of matrices from $$2 \times 2$$ matrices to arbitrary dimension matrices. The properties are quite similar, but there are a number of important ideas that need to be mastered to handle these more general cases. The concepts include eigenvalues and eigenvectors of general matrices, completeness and orthonormality, trace, partial trace, determinant, positive semidefinite matrices, and functions of matrices. We also discuss two other advanced mathematics ideas related to summations and counting in the final sections of the chapter, ending with a motivation for the Born rule.

DOI
10.1007/978-3-031-87845-9_13

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This chapter introduces the essential topics and to show how eigenvalues and eigenvectors are calculated. In quantum computation it is relevant to consider cases in which the eigenvalues are all real and the eigenvectors are orthogonal. In the study of quantum computation, the matrices with which we usually work have properties that enable us to be quite specific about the nature of the eigenvalues and eigenvectors. Post-measurement the system is in a state represented by an eigenvector or eigenvectors. In quantum computation it is relevant to consider cases in which the eigenvalues are all real and the eigenvectors are orthogonal.
The quantities in physics that can be measured are conventionally denoted observables : entities that can be observed (directly or indirectly) and to which a value in appropriate units can be assigned. Examples include position, momentum, and energy. Unlike in classical physics, the concept of an observable in quantum theory possesses a significance beyond that of the measurable: to each observable there corresponds a unique operator in an abstract, linear vector space. These operators (and others) govern all aspects of quantum systems. Quantum operators act on abstract vectors in the linear space. Especially important are the eigenvectors and eigenvalues of these operators, in particular because some of the eigenvectors are the quantum states mentioned in the preceding chapters. Furthermore, for any operator that is the mathematical image of an observable, its eigenvalues constitute the entire set of values that measurement of the observable can yield . Finally, from the quantum states one can obtain the probabilities for the occurrence of events in the microscopic world, for example, the most likely value that measurement of an observable will yield. The concepts of operators and eigenvalue problems, as well as those of quantal probabilities, of Hermiticity, and of complete sets, were encountered and exemplified in the preceding chapter using the language of differential equations. While much of quantum mechanics could be presented in terms of this language alone, certain fundamental concepts (spin, for instance) cannot be, and it eventually becomes necessary to consider quantum mechanics as a theoretical framework embedded in that abstract linear vector space known as a Hilbert space.