Lecture

Cholesky Factorization: Theory and Algorithm

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Description

This lecture covers the Cholesky factorization method for symmetric positive definite matrices, explaining the theory behind it and the algorithm to compute it efficiently. The Cholesky factorization is a crucial tool in numerical linear algebra, providing a unique decomposition of a matrix into a product of a lower triangular matrix and its transpose.

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Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Matrix decomposition

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.

Cholesky 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.

LU decomposition

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.

QR decomposition

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning ) and R is an upper triangular matrix (also called right triangular matrix).

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

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