In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... A matrix function usually denotes a function which maps a matrix to a matrix. ... For the square matrix section, see square matrix. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

Let X be an n×n real or complex matrix. The exponential of X, denoted by e^X or exp(X), is the n×n matrix given by the power series In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

The above series always converges, so the exponential of X is well-defined. Note that if X is a 1×1 matrix the matrix exponential of X corresponds with the ordinary exponential of X thought of as a number.

Properties

Let X and Y be n×n complex matrices and let a and b be arbitrary complex numbers. We denote the n×n identity matrix by I and the zero matrix by 0. The matrix exponential satisfies the following properties: In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... In mathematics, a zero matrix is a matrix with all its entries being zero. ...

• If then
• If Y is invertible then
• det(e^X) = e^tr(X), where tr(X) is the trace of X.
• exp(X^T) = (exp X)^T, where X^T denotes the transpose of X. It follows that if X is symmetric then e^X is also symmetric, and that if X is skew-symmetric then e^X is orthogonal.
• exp(X^*) = (exp X)^*, where X^* denotes the conjugate transpose of X. It follows that if X is Hermitian then e^X is also Hermitian, and that if X is skew-Hermitian then e^X is unitary.

In linear algebra, an n-by-n (square) matrix is called invertible or non-singular 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. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or A′) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ... In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A* is also its negative. ... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...

Linear differential equations

One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations. Indeed, it follows from equation (1) below that the solution of In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. ...

where A is a matrix, is given by

The matrix exponential can also be used to solve the inhomogeneous equation

See the section on applications below for examples.

There is no closed-form solution for differential equations of the form

where A is not constant, but the Magnus series gives the solution as an infinite sum. Magnus series refers to specific equations used in computational mathematics. ...

The exponential of sums

We know that the exponential function satisfies e^{x + y} = e^xe^y for any numbers x and y. The same goes for commuting matrices: If the matrices X and Y commute (meaning that XY = YX), then

However, if they do not commute, then the above equality does not necessarily hold. In that case, we can use the Baker-Campbell-Hausdorff formula to compute e^{X + Y}. In mathematics, the Baker-Campbell-Hausdorff formula is the solution to for non-commuting x and y. ...

The exponential map

Note that the exponential of a matrix is always a non-singular matrix. The inverse of e^X is given by e^−X. This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map 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. ... 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. ...

from the space of all n×n matrices to the general linear group, i.e. the group of all non-singular matrices. In fact, this map is surjective which means that every non-singular matrix can be written as the exponential of some other matrix (for this, it is essential to consider the field C of complex numbers and not R). The matrix logarithm gives an inverse to this map. In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ... This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, the logarithm of a matrix is a generalization of the scalar logarithm to matrices. ...

For any two matrices X and Y, we have

where || · || denotes an arbitrary matrix norm. It follows that the exponential map is continuous and Lipschitz continuous on compact subsets of M_n(C). In mathematics, the term Matrix Norm can have two meanings: A vector norm on matrices, i. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a function f : M → N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) ≤ K d(x, y) for all x and y... In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ...

The map

defines a smooth curve in the general linear group which passes through the identity element at t = 0. In fact, this gives a one-parameter subgroup of the general linear group since Smooth could mean many things, including: Smooth function, a function that is infinitely differentiable, used in calculus and topology. ... In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is...

The derivative of this curve (or tangent vector) at a point t is given by In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...

The derivative at t = 0 is just the matrix X, which is to say that X generates this one-parameter subgroup.

More generally,

Computing the matrix exponential

Finding reliable and accurate methods to compute the matrix exponential is difficult, and this is still a topic of considerable current research in mathematics and numerical analysis. Some methods are listed below.

Diagonalizable case

If a matrix is diagonal: In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ...

then its exponential can be obtained by just exponentiating every entry on the main diagonal:

This also allows one to exponentiate diagonalizable matrices. If A = UDU⁻¹ and D is diagonal, then e^A = Ue^DU⁻¹. Application of Sylvester's formula yields the same result. In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ... In matrix theory, Sylvesters formula, named after James Joseph Sylvester, expresses matrix functions in terms of the eigenvalues and eigenvectors of a matrix. ...

Nilpotent case

A matrix N is nilpotent if N^q = 0 for some integer q. In this case, the matrix exponential e^N can be computed directly from the series expansion, as the series terminates after a finite number of terms: In mathematics, a nilpotent matrix is an n×n square matrix M such that for some positive integer q. ...

General case

An arbitrary matrix X (over an algebraically closed field) can be expressed uniquely as sum In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ...

where

• A is diagonalizable
• N is nilpotent
• A commutes with N (i.e. AN = NA)

This means we can compute the exponential of X by reducing to the previous two cases:

Note that we need the commutativity of A and N for the last step to work.

Another (closely related) method is to work with the Jordan form of X. Suppose J is the Jordan form of X, with P the transition matrix. Then 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, to what extent M can be simplified into a standard...

Also, since

Therefore, we need only know how to compute the matrix exponential of a Jordan block. But each Jordan block is of the form

where N is a special nilpotent matrix. The matrix exponential of this block is given by

Calculations

Suppose that we want to compute the exponential of

Its Jordan form is

where the transition matrix is given by

Let us first calculate exp(J). We have

The exponential of a 1×1 matrix is just the exponential of the one entry of the matrix, so exp(J₁(4)) = [e⁴]. The exponential of J₂(16) can be calculated by the formula exp(λI+N) = e^λ exp(N) mentioned above; this yields

Therefore, the exponential of the original matrix B is

Applications

Linear differential equations

The matrix exponential has applications to systems of linear differential equations. Recall from earlier in this article that a differential equation of the form In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ...

y′ = Cy

has solution e^Cxy(0). If we consider the vector

we can express a system of coupled linear differential equations as

If we make an ansatz and use an integrating factor of e^−Ax and multiply throughout, we obtain Ansatz (Ger. ...

If we can calculate e^Ax, then we can obtain the solution to the system.

Example (homogeneous)

Say we have the system

We have the associated matrix

In the example above, we have calculated the matrix exponential

so the general solution of the system is

that is,

Inhomogeneous case - variation of parameters

For the inhomogeneous case, we can use a method akin to variation of parameters. We seek a particular solution of the form y_p(t) = exp(tA)z(t) : In mathematics, variation of parameters is a technique used in solving certain second order linear inhomogeneous ordinary differential equations. ...

For y_p to be a solution:

So,

where c is determined by the initial conditions of the problem.

Example (inhomogeneous)

Say we have the system

So we then have

and

From before, we have the general solution to the homogeneous equation, Since the sum of the homogeneous and particular solutions give the general solution to the inhomogeneous problem, now we only need to find the particular solution (via variation of parameters).

We have, above:

which can be further simplified to get the requisite particular solution determined through variation of parameters.

See also

• Matrix logarithm
• exponential function
• exponential map
• vector flow
• Golden-Thompson inequality
• Phase-type distribution

In mathematics, the logarithm of a matrix is a generalization of the scalar logarithm to matrices. ... The exponential function is one of the most important functions in mathematics. ... There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ... In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. ... In mathematics, the Golden-Thompson inequality is as follows. ... A phase-type distribution is a probability distribution that results from a system one or more inter-related poisson processes occurring in sequence, or phases. ...

References

• Roger A. Horn and Charles R. Johnson. Topics in Matrix Analysis. Cambridge University Press, 1991. ISBN 0-521-46713-6.
• C. Moler and C. Van Loan. Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review 45, 3-49 (2003)

External links

• Eric W. Weisstein, Matrix Exponential at MathWorld.
• Module for the Matrix Exponential