Error Gauss_elimination Singular Matrix
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Gaussian Elimination Singular Matrix
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on 13 June 2012 by John Someone asked me on TwitterIs there a trick to make an singular (non-invertible) matrix invertible?The only response I could think of in less than 140 characters wasDepends on what you're trying to accomplish.Here gauss elimination method example I'll give a longer explanation.So, can you change a singular matrix
Nonsingular Matrix
just a little to make it non-singular? Yes, and in fact you can change it by as singular value decomposition little as you'd like. Singular square matrices are an infinitely thin subset of the space of all square matrices, and any tiny random perturbation is almost certain to http://www.cfd-online.com/Forums/fluent/40391-gauss_elimination-singular-matrix.html take you out of that thin subset. (“Almost certain” is actually a technical term here, meaning with probability 1.)Adding a tiny bit of noise to a singular matrix makes it non-singular. But why did you want a non-singular matrix? Presumably to solve some system of equations. A little noise makes the system go from theoretically impossible to http://www.johndcook.com/blog/2012/06/13/matrix-condition-number/ solve to theoretically possible to solve, but that may not be very useful in practice. It will let you compute a solution, but that solution may be meaningless.A little noise makes the determinant go from zero to near zero. But here is an important rule in numerical computation:If something is impossible at 0, it will be hard near zero."Hard" could mean difficult to do or it could mean the solution is ill-behaved.Instead of asking whether a matrix is invertible, let's ask how easy it is to invert. That is, let's change a yes/no question into a question with a continuum of answers. The condition number of a matrix measures how easy the matrix is to invert. A matrix that is easy to invert has a small condition number. The harder it is to invert a matrix, the larger its condition number. A singular matrix is infinitely hard to invert, and so it has infinite condition number. A small perturbation of a singular matrix is non-singular, but the condit
från GoogleLogga inDolda fältBöckerbooks.google.se - Designed for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and https://books.google.com/books?id=NiYg4iU8lWkC&pg=PA512&lpg=PA512&dq=error+gauss_elimination+singular+matrix&source=bl&ots=3piRKHZchv&sig=Yan61DhHNLTVhS74ysuDicAsx68&hl=en&sa=X&ved=0ahUKEwjthqGf_cvPAhVGbR4KHeMAAzQQ6AEIX practical application of linear algebra and functional analysis. The book https://books.google.com/books?id=PlUWqQzY-4EC&pg=PA251&lpg=PA251&dq=error+gauss_elimination+singular+matrix&source=bl&ots=MdSHubDO2E&sig=v2VRv5HW52Bj_2smaNwvzcpouGw&hl=en&sa=X&ved=0ahUKEwjthqGf_cvPAhVGbR4KHeMAAzQQ6AEIY is self-contained, beginning with elementary principles, basic concepts,...https://books.google.se/books/about/Linear_Algebra_and_Linear_Operators_in_E.html?hl=sv&id=NiYg4iU8lWkC&utm_source=gb-gplus-shareLinear Algebra and Linear Operators in EngineeringMitt bibliotekHjälpAvancerad boksökningKöp e-bok – 1 176,43 krSkaffa ett tryckt exemplar av den här bokenAccess Online via ElsevierAmazon.co.ukAdlibrisAkademibokandelnBokus.seHitta boken i ett bibliotekAlla försäljare»Linear Algebra and Linear Operators in singular matrix Engineering: With Applications in Mathematica®H. Ted Davis, Kendall T. ThomsonAcademic Press, 12 juli 2000 - 547 sidor 0 Recensionerhttps://books.google.se/books/about/Linear_Algebra_and_Linear_Operators_in_E.html?hl=sv&id=NiYg4iU8lWkCDesigned for advanced engineering, physical science, and applied mathematics students, this innovative textbook is an introduction to both the theory and practical application of linear algebra and functional analysis. The book gauss elimination method is self-contained, beginning with elementary principles, basic concepts, and definitions. The important theorems of the subject are covered and effective application tools are developed, working up to a thorough treatment of eigenanalysis and the spectral resolution theorem. Building on a fundamental understanding of finite vector spaces, infinite dimensional Hilbert spaces are introduced from analogy. Wherever possible, theorems and definitions from matrix theory are called upon to drive the analogy home. The result is a clear and intuitive segue to functional analysis, culminating in a practical introduction to the functional theory of integral and differential operators. Numerous examples, problems, and illustrations highlight applications from all over engineering and the physical sciences. Also included are several numerical applications, complete with Mathematica solutions and code, giving the student a "hands-on" introduction to numerical analysis. Linear Algebra and Linear Operators in Engineering is ideally suited as the main text o
från GoogleLogga inDolda fältBöckerbooks.google.se - This book distinguishes itself from the many other textbooks on the topic of linear algebra by including mathematical and computational chapters along with examples and exercises with Matlab. In recent years, the use of computers in many areas of engineering and science has made it essential for students...https://books.google.se/books/about/Numerical_Linear_Algebra.html?hl=sv&id=PlUWqQzY-4EC&utm_source=gb-gplus-shareNumerical Linear AlgebraMitt bibliotekHjälpAvancerad boksökningKöp e-bok – 378,62 krSkaffa ett tryckt exemplar av den här bokenSpringer ShopAmazon.co.ukAdlibrisAkademibokandelnBokus.seHitta boken i ett bibliotekAlla försäljare»Numerical Linear AlgebraGrégoire Allaire, Sidi Mahmoud KaberSpringer Science & Business Media, 17 dec. 2008 - 271 sidor 1 Recensionhttps://books.google.se/books/about/Numerical_Linear_Algebra.html?hl=sv&id=PlUWqQzY-4ECThis book distinguishes itself from the many other textbooks on the topic of linear algebra by including mathematical and computational chapters along with examples and exercises with Matlab. In recent years, the use of computers in many areas of engineering and science has made it essential for students to get training in numerical methods and computer programming. Here, the authors use both Matlab and SciLab software as well as covering core standard material. It is intended for libraries; scientists and researchers; pharmaceutical industry. Förhandsvisa den här boken » Så tycker andra-Skriv en recensionVi kunde inte hitta några recensioner.Utvalda sidorTitelsidaIndexReferensInnehållIntroduction1 12 Least Squares Fitting4 13 Vibrations of a Mechanical System8 14 The Vibrating String10 15 Image Compression by the SVD Factorization12 Definition and Properties of Matrices15 22 Matrices17 221 Trace and Determinant19 72 Main Results126 73 Numerical Algorithms128 732 Normal Equation Method131 733 QR Factorization Method132 734 Householder Algorithm136 74 Exercises140 Simple Iterative Methods143 82 Jacobi GaussSeidel and Relaxation Methods147 Mer222 Special Matrices20 223 Rows and Columns21 224 Row and Column Permutation22 23 Spectral Theory of Matrices23 24 Matrix Triangularization26 25 Matrix Diagonalization28 26 MinMax Principle31 27 Singular Values of a Matr