Titlebook: Linear Algebra; J. H. Wilkinson,C. Reinsch,F. L. Bauer (Chief edit Book 1971 Springer-Verlag Berlin Heidelberg 1971 algebra.linear algebra

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The , and , Algorithms for Symmetric Matricesis unitary and . is upper-triangular then . that is, . is unitarily similar to . By repeated application of the above result a sequence of matrices which are unitarily similar to a given matrix .. may be derived from the relations . and, in general, .. tends to upper-triangular form.

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Overview: 978-3-662-38854-9978-3-662-39778-7

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Symmetric Decomposition of a Positive Definite MatrixThe methods are based on the following theorem due to . [1].

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Inversion of Positive Definite Matrices by the Gauss-Jordan MethodLet . be a real . matrix and . the induced mapping ..... If .. ≠ 0, then one can solve the first of these equations for .. and insert the result into the remaining equations. This leads to the system . with ..

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Solution of Real and Complex Systems of Linear EquationsIf . is a non-singular matrix then, in general, it can be factorized in the form ., where . is lower-triangular and . is upper-triangular. The factorization, when it exists, is unique to within a non-singular diagonal multiplying factor.

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Elimination with Weighted Row Combinations for Solving Linear Equations and Least Squares ProblemsLet . be a matrix of . rows and . columns, . and only if the columns are linearly independent, then for any vector . there exists a unique vector . minimizing the Euclidean norm of ..