**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.

Lecture# Linear Systems: Chapters 4, 5, 6

Description

This lecture covers the relationship between linear systems and optimization, focusing on elimination and LU decomposition. It explores the link between symmetric positive definite matrices and optimization problems, with a detailed explanation of the concepts.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

In course

MATH-251(b): Numerical analysis

The students will learn key numerical techniques for solving standard mathematical problems in science and engineering. The underlying mathematical theory and properties are discussed.

Instructors (2)

Related concepts (91)

Definite matrix

In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector where is the transpose of . More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector where denotes the conjugate transpose of Positive semi-definite matrices are defined similarly, except that the scalars and are required to be positive or zero (that is, nonnegative).

Inner product space

In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates.

Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to The matrix is skew-symmetric because Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2.

Quadratic form

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.

Bilinear form

In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The dot product on is an example of a bilinear form.

Related lectures (675)

Linear systems resolutionMATH-212: Analyse numérique et optimisation

Covers the resolution of linear systems and its link to optimization problems.

Singular Value Decomposition: Theoretical FoundationsMATH-111(e): Linear Algebra

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

Numerical Analysis: Linear SystemsMATH-251(c): Numerical analysis

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

Singular Value Decomposition: Applications and InterpretationMATH-111(e): Linear Algebra

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

Optimal Control: KKT ConditionsMATH-351: Advanced numerical analysis

Explores optimal control and KKT conditions for non-linear optimization with constraints.