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Lecture# Linear Systems: Factorization and Cholesky

Description

This lecture covers the uniqueness of solutions in linear systems, the Cholesky factorization, LU factorization, and the cost of factorization. It also discusses the existence of solutions, regular matrices, and the formulation of linear systems. The instructor explains the importance of regular matrices, the factorization of Cholesky, and the cost of LU factorization. Additionally, the lecture delves into the cost of resolution, the main minors of matrices, and the precision of Hilbert matrices. The Sylvester criterion, iterative methods, and the resolution of systems using factorization are also explored.

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Instructors (2)

Related concepts (109)

MATH-250: Numerical analysis

Construction and analysis of numerical methods for the solution of problems from linear algebra, integration, approximation, and differentiation.

Related lectures (57)

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.

In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ʃəˈlɛski ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices, and posthumously published in 1924. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.

In numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.

In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by the ordered triple since it makes all three equations valid. The word "system" indicates that the equations should be considered collectively, rather than individually.

In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.

Covers the formulation of linear systems, direct and iterative methods for solving them, and the cost of LU factorization.

Explains the construction of U, verification of results, and interpretation of SVD in matrix decomposition.

Covers the formulation of linear systems and iterative methods like Richardson, Jacobi, and Gauss-Seidel.

Covers the analysis of linear systems, focusing on methods such as Jacobi and Richardson for solving linear equations.

Covers the theoretical foundations of Singular Value Decomposition, explaining the decomposition of a matrix into singular values and vectors.