Titlebook: Numerical Linear Algebra and Matrix Factorizations; Tom Lyche Textbook 2020 Springer Nature Switzerland AG 2020 numerical linear algebra.m

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The Kronecker ProductMatrices arising from 2D and 3D problems sometimes have a Kronecker product structure. Identifying a Kronecker structure can be very rewarding since it simplifies the derivation of properties of such matrices.

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The Singular Value Decompositionpplications include solving over-determined equations, principal component analysis in statistics, numerical determination of the rank of a matrix, algorithms used in search engines, and the theory of matrices.

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Orthonormal and Unitary Transformationsngular form and more generally a rectangular matrix to upper triangular (also called upper trapezoidal) form. This lead to a decomposition of a rectangular matrix known as a . and a reduced form which we refer to as a .. The QR decomposition and factorization will be used in later chapters to solve least squares- and eigenvalue problems.

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Least Squaresay a .-norm, and look for . which minimizes .. The use of the one and . norm can be formulated as linear programming problems, while the Euclidian norm leads to a linear system and has applications in statistics. Only this norm is considered here.

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The Conjugate Gradient Methodo not have to estimate any such parameter. However the conjugate gradient method is restricted to positive definite systems. We also consider a mathematical formulation of the .. It is used to speed up convergence of the conjugate gradient method. We only give one example of a possible preconditioner.