Abstract
The numerical evaluation of the inverse, determinant, eigenvalues, and eigenvectors of a square matrix \(\mathbf {A}\) is considered. The reasons why inverting a matrix or computing its determinant is seldom useful are explained. Efficient methods for doing so when it does turn out to be necessary are presented. Examples illustrate the variety of the problems requiring the computation of eigenvalues and eigenvectors. Approaches best avoided for doing so are pointed out, and readily available alternatives are described, which may be used to compute a single eigenvalue of interest and its eigenvector (as in the PageRank algorithm used by Google to order its answers to a given query) or all of the eigenvalues (as when computing all the roots of a polynomial equation). A method computing all the eigenvalues of a matrix can also be used to compute its eigenvectors, which is particularly easy if the matrix is symmetric.
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Walter, É. (2014). Solving Other Problems in Linear Algebra. In: Numerical Methods and Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-07671-3_4
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DOI: https://doi.org/10.1007/978-3-319-07671-3_4
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