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Encyclopedia > Matrix logarithm

In mathematics, the logarithm of a matrix is a generalization of the scalar logarithm to matrices. It is in some sense an inverse function of the matrix exponential. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Above is the graph plots of Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary exponential function. ...

Contents

Definition

A matrix B is a logarithm of a given matrix A if the matrix exponential of B is A: In mathematics, the matrix exponential is a function on square matrices analogous to the ordinary exponential function. ...

Properties

A matrix has a logarithm if and only if it is invertible. However, this logarithm may be complex even if all the entries in the matrix are real numbers. In any case, the logarithm is not unique. It has been suggested that this article or section be merged with Logical biconditional. ... In linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...


Calculating the logarithm of a diagonalizable matrix

A method for finding ln A for a diagonalizable matrix A is the following: In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...

Find the matrix V of eigenvectors of A (each column of V is an eigenvector of A).
Find the inverse V−1 of V.
Let
Then A′ will be a diagonal matrix whose diagonal elements are eigenvalues of A.
Replace each diagonal element of A′ by its (natural) logarithm in order to obtain lnA'.
Then

That the logarithm of A might be a complex matrix even if A is real then follows from the fact that a matrix with real entries might nevertheless have complex eigenvalues (this is true for example for rotation matrices). The non-uniqueness of the logarithm of a matrix follows from the non-uniqueness of the logarithm of a complex number. In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... A rotation matrix is a matrix which when multiplied by a vector has the effect of changing the direction of the vector but not its magnitude. ...


The logarithm of a non-diagonalizable matrix

The algorithm illustrated above does not work for non-diagonalizable matrices, such as

For such matrices one needs to find its Jordan decomposition and, rather than computing the logarithm of diagonal entries as above, one would calculate the logarithm of the Jordan blocks. In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K containing the eigenvalues of M, to what extent can M...


The latter is accomplished by noticing that one can write a Jordan block as

where K is a matrix with zeros on and under the main diagonal. (The number λ is nonzero by the assumption that the matrix whose logarithm one attempts to take is invertible.)


Then, by the formula

one gets

This series in general does not converge for any matrix K, as it would not for any real number with absolute value greater than unity, however, this particular K is a nilpotent matrix, so the series actually has a finite number of terms (Km is zero if m is the dimension of K). In mathematics, a series is often represented as the sum of a sequence of terms. ... In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...


Using this approach one finds

A functional analysis perspective

A square matrix represents a linear operator on the Euclidean space Rn where n is the dimension of the matrix. Since such a space is finite-dimensional, this operator is actually bounded. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...


Using the tools of holomorphic functional calculus, given a holomorphic function f(z) defined on an open set in the complex plane and a bounded linear operator T, one can calculate f(T) as long as f(z) is defined on the spectrum of T. In mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ...


The function f(z)=ln z can be defined on any simply connected open set in the complex plane not containing the origin, and it is holomorphic on such a domain. This implies that one can define ln T as long as the spectrum of T does not contain the origin and there is a path going from the origin to infinity not crossing the spectrum of T (as such, if the spectrum of T is a circle with the origin inside of it, it is impossible to define ln T). A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...


Back to the particular case of an Euclidean space, the spectrum of a linear operator on this space is the set of eigenvalues of its matrix, and so is a finite set. As long as the origin is not in the spectrum (the matrix is invertible), one obviously satisfies the path condition from the previous paragraph, and such, the theory implies that ln T is well-defined. The non-uniqueness of the matrix logarithm then follows from the fact that one can choose more than one branch of the logarithm which is defined on the set of eigenvalues of a matrix.


See also


 
 

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