Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Precision Problems: Hilbert Matrix

In courses (2)

MATH-250: Advanced numerical analysis I

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

MATH-251(c): Numerical analysis

Le cours présente des méthodes numériques pour la résolution de problèmes mathématiques comme des systèmes d'équations linéaires ou non linéaires, l'approximation de fonctions, l'intégration et la dérivation, et les équations différentielles. ...

Description

This lecture covers precision problems in linear systems, focusing on the conditioning of matrices and the accuracy of solutions obtained through direct methods. The instructor explains the concept of conditioning, the impact on solution accuracy, and the limitations of direct methods when dealing with ill-conditioned matrices.

Instructors (3)

Annalisa Buffa

Simone Deparis

Daniel Kressner

Official source

Related concepts (14)

Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular, nondegenerate or (rarely used) regular), if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix B is uniquely determined by A, and is called the (multiplicative) inverse of A, denoted by A−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is , while an example of a 3×3 diagonal matrix is. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size).

Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.

Matrix multiplication

In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number of rows of the first and the number of columns of the second matrix. The product of matrices A and B is denoted as AB.

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.

Related lectures (42)

Linear Algebra: Matrices Properties

Explores properties of 3x3 matrices with real coefficients and determinant calculation methods.

Vectorization in Python: Efficient Computation with Numpy

Covers vectorization in Python using Numpy for efficient scientific computing, emphasizing the benefits of avoiding for loops and demonstrating practical applications.

Matrix of Cofactors and Inverse Matrix Formula MOOC: Linear Algebra (Part 2)

Covers the concept of the matrix of cofactors and a formula to calculate the inverse of a matrix.

Linear Independence and Basis

Explains linear independence, basis, and matrix rank with examples and exercises.

Linear Algebra: Reduction of Linear Application