Logarithm of a matrix

In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in an element of a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra. The exponential of a matrix A is defined by Given a matrix B, another matrix A is said to be a matrix logarithm of B if eA = B. Because the exponential function is not bijective for complex numbers (e.g. ), numbers can have multiple complex logarithms, and as a consequence of this, some matrices may have more than one logarithm, as explained below. If B is sufficiently close to the identity matrix, then a logarithm of B may be computed by means of the following power series: Specifically, if , then the preceding series converges and . The rotations in the plane give a simple example. A rotation of angle α around the origin is represented by the 2×2-matrix For any integer n, the matrix is a logarithm of A. ⇔ where ... qed. Thus, the matrix A has infinitely many logarithms. This corresponds to the fact that the rotation angle is only determined up to multiples of 2π. In the language of Lie theory, the rotation matrices A are elements of the Lie group SO(2). The corresponding logarithms B are elements of the Lie algebra so(2), which consists of all skew-symmetric matrices. The matrix is a generator of the Lie algebra so(2). The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting. A complex matrix has a logarithm if and only if it is invertible. The logarithm is not unique, but if a matrix has no negative real eigenvalues, then there is a unique logarithm that has eigenvalues all lying in the strip .

About this result

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.

Related concepts (12)

Related courses (16)

Related lectures (207)

MATH-111(e): Linear Algebra

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

MATH-453: Computational linear algebra

This course provides an overview of advanced techniques for solving large-scale linear algebra problems, as they typically arise in applications. A central goal of this course is to give the ability t

MATH-429: Lie groups

We will discuss the basic structure of Lie groups and of their associated Lie algebras along with their finite dimensional representations and with a special emphasis on matrix Lie groups.

Logarithm of a matrix

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.

Analytic function of a matrix

In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations. There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained. All of the following techniques yield the same matrix function, but the domains on which the function is defined may differ.

Understanding Vacuum Energy in Inflation

Explores vacuum energy during inflation and the dynamics of scalar and vector fields, emphasizing the importance of seeking clarification and providing details about upcoming exams.

Diagonalization of Matrices MATH-111(e): Linear Algebra

Explores the diagonalization of matrices through eigenvalues and eigenvectors, emphasizing the importance of bases and subspaces.

Science-fiction: Hybridity HUM-219: Science and literature C

Delves into hybridity in science-fiction, exploring space opera and science-fantasy themes like hard science and cyberpunk.