Lecture

Linear Algebra Review: Convex Optimization

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In course

MGT-418: Convex optimization

This course introduces the theory and application of modern convex optimization from an engineering perspective. ...

Description

This lecture provides a comprehensive review of linear algebra concepts essential for understanding convex optimization. Topics covered include vector norms, eigenvalue decomposition, singular value decomposition, matrix norms, and properties of matrices such as trace and determinant. The lecture also delves into convex functions, inequalities involving vector norms, and the rule of Sarrus for 2x2 and 3x3 matrices. Additionally, it explores the concepts of positive semidefinite and positive definite matrices, principal minors, and the square root of positive semidefinite matrices. The instructor, Bahar Taskesen, presents various examples and proofs to solidify the understanding of these fundamental linear algebra concepts.

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Instructor

Daniel Kuhn

Official source

Ontological neighbourhood

Mathematics

Algebra: Linear algebra

Related concepts (32)

Singular value decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an complex matrix M is a factorization of the form where U is an complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, V is an complex unitary matrix, and is the conjugate transpose of V.

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).

Convex optimization

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.

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.

Matrix norm

In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions). Given a field of either real or complex numbers, let be the K-vector space of matrices with rows and columns and entries in the field . A matrix norm is a norm on . This article will always write such norms with double vertical bars (like so: ).

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