Matrix exponential - Citizendia

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A matrix function usually denotes a function which maps a matrix to a matrix In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Let X be an n×n real or complex matrix. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally The exponential of X, denoted by e^X or exp(X), is the n×n matrix given by the power series

$e^X = \sum_{k=0}^\infty{1 \over k!}X^k.$

The above series always converges, so the exponential of X is well-defined. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + Note that if X is a 1×1 matrix the matrix exponential of X is a 1×1 matrix consisting of the ordinary exponential of the single element of X.

1 Properties
2 Computing the matrix exponential
3 Calculations
4 Applications
- 4.1 Linear differential equations
5 See also
6 References
7 External links

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. In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main In Mathematics, particularly Linear algebra, a zero matrix is a matrix with all its entries being zero. The matrix exponential satisfies the following properties:

e⁰ = I.
e^aXe^bX = e⁽^{a + b)X}.
e^Xe^−X = I.
If XY = YX then e^Xe^Y = e^Ye^X.
If Y is invertible then e^YXY⁻¹ = Ye^XY⁻¹. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by-
det(e^X) = e^tr(X), where tr(X) is the trace of X. 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
exp(X^T) = (exp X)^T, where X^T denotes the transpose of X. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a 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. In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} In Linear algebra, a skew-symmetric (or antisymmetric) matrix is a Square matrix A whose Transpose is also its negative In Matrix theory, a real orthogonal matrix is a square matrix Q whose Transpose is its inverse: Q^T
exp(X*) = (exp X)*, where X* denotes the conjugate transpose of X. In Mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m -by- n matrix A with 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. A Hermitian matrix (or self-adjoint matrix) is a Square matrix with complex entries which is equal to its own Conjugate transpose &mdash that In Linear algebra, a Square matrix (or more generally a linear transformation from a complex Vector space with a Sesquilinear norm to itself In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^*

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. In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its Indeed, it follows from equation (1) below that the solution of

$\frac{d}{dt} y(t) = Ay(t), \quad y(0) = y_0,$

where A is a matrix, is given by

$y(t) = e^{At} y_0. \,$

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

$\frac{d}{dt} y(t) = Ay(t) + z(t), \quad y(0) = y_0.$

See the section on applications below for examples.

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

$\frac{d}{dt} y(t) = A(t) \, y(t), \quad y(0) = y_0,$

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 real numbers (scalars) x and y. The same goes for commuting matrices: If the matrices X and Y commute (meaning that XY = YX), then

$e^{X+Y} = e^Xe^Y. \,$

However, if they do not commute, then the above equality does not necessarily hold. In which 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 Z = \log(e^X e^Y\ for non- commuting X

The exponential map

Note that the exponential of a matrix is always a non-singular matrix. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- The inverse of e^X is given by e^−X. In Linear algebra, an n -by- n (square matrix A is called invertible or non-singular if there exists an n -by- This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map

$\exp \colon M_n(\mathbb C) \to \mbox{GL}(n,\mathbb C)$

from the space of all n×n matrices to the general linear group, i. In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation e. the group of all non-singular matrices. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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). In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every The matrix logarithm gives an inverse to this map. In Mathematics, the logarithm of a matrix is a Matrix function which generalizes the scalar Logarithm to matrices.

For any two matrices X and Y, we have

$\| e^{X+Y} - e^X \| \le \|Y\| e^{\|X\|} e^{\|Y\|},$

where || · || denotes an arbitrary matrix norm. In Mathematics, a matrix norm is a natural extension of the notion of a Vector norm to matrices. It follows that the exponential map is continuous and Lipschitz continuous on compact subsets of M_n(C). In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions

The map

$t \mapsto e^{tX}, \qquad t \in \mathbb R$

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

$e^{tX}e^{sX} = e^{(t+s)X}.\,$

The derivative of this curve (or tangent vector) at a point t is given by

$\frac{d}{dt}e^{tX} = Xe^{tX}. \qquad (1)$

The derivative at t = 0 is just the matrix X, which is to say that X generates this one-parameter subgroup. In Mathematics, a one-parameter group or one-parameter subgroup usually means a continuous Group homomorphism φ: R

More generally,

$\frac{d}{dt}e^{X(t)} = \int_0^1 e^{(1-\alpha) X(t)} \frac{dX(t)}{dt} e^{\alpha X(t)}\,d\alpha.$

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:

$A=\begin{bmatrix} a_1 & 0 & \ldots & 0 \\ 0 & a_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & a_n \end{bmatrix},$

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

$e^A=\begin{bmatrix} e^{a_1} & 0 & \ldots & 0 \\ 0 & e^{a_2} & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & e^{a_n} \end{bmatrix}.$

This also allows one to exponentiate diagonalizable matrices. In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero In Linear algebra, a Square matrix A is called diagonalizable if it is similar to a Diagonal matrix, i If $A = U D U - 1$ and D is diagonal, then $e A = U e D U - 1$ . Application of Sylvester's formula yields the same result. In Matrix theory, Sylvester's formula, named after James Joseph Sylvester, expresses matrix functions in terms of the eigenvalues and eigenvectors of a

Nilpotent case

A matrix N is nilpotent if N^q = 0 for some integer q. In Mathematics, a nilpotent matrix is an n × n Square matrix M such that M^q = 0\ for some positive 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:

$e^N = I + N + \frac{1}{2}N^2 + \frac{1}{6}N^3 + \cdots + \frac{1}{(q-1)!}N^{q-1}.$

Generalization

When the minimal polynomial of a matrix X can be factored into a product of first degree polynomials, it can be expressed as a sum

$X = A + N \,$

where

A is diagonalizable
N is nilpotent
A commutes with N (i. In Linear algebra, the minimal polynomial of an n -by- n matrix A over a field F is the Monic polynomial e. AN = NA)

This is the Dunford decomposition.

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

$e^X = e^{A+N} = e^A e^N. \,$

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

Another (closely related) method if the field is algebraically closed is to work with the Jordan form of X. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Linear algebra, Jordan normal form (often called Jordan canonical form)shows that a given square matrix M over a field K Suppose that X = PJP⁻¹ where J is the Jordan form of X. Then

$e^{X}=Pe^{J}P^{-1}.\,$

Also, since

$J=J_{a_1}(\lambda_1)\oplus J_{a_2}(\lambda_2)\oplus\cdots\oplus J_{a_n}(\lambda_n),$

$\begin{align} e^{J} & {} = \exp \big( J_{a_1}(\lambda_1)\oplus J_{a_2}(\lambda_2)\oplus\cdots\oplus J_{a_n}(\lambda_n) \big) \\ & {} = \exp \big( J_{a_1}(\lambda_1) \big) \oplus \exp \big( J_{a_2}(\lambda_2) \big) \oplus\cdots\oplus \exp \big( J_{a_k}(\lambda_k) \big). \end{align}$

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

$J_{a}(\lambda) = \lambda I + N \,$

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

$e^{\lambda I + N} = e^{\lambda}e^N. \,$

Calculations

Suppose that we want to compute the exponential of

$B=\begin{bmatrix} 21 & 17 & 6 \\ -5 & -1 & -6 \\ 4 & 4 & 16 \end{bmatrix}.$

Its Jordan form is

$J = P^{-1}BP = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 16 & 1 \\ 0 & 0 & 16 \end{bmatrix},$

where the matrix P is given by

$P=\begin{bmatrix} -\frac14 & 2 & \frac54 \\ \frac14 & -2 & -\frac14 \\ 0 & 4 & 0 \end{bmatrix}.$

Let us first calculate exp(J). We have

$J=J_1(4)\oplus J_2(16)$

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 $J 2 (16)$ can be calculated by the formula exp(λ $I$ +N) = e^λ exp(N) mentioned above; this yields

$\exp \left( \begin{bmatrix} 16 & 1 \\ 0 & 16 \end{bmatrix} \right) = e^{16} \exp \left( \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \right) = e^{16} \left(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} + {1 \over 2!}\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} + \cdots \right) = \begin{bmatrix} e^{16} & e^{16} \\ 0 & e^{16} \end{bmatrix}.$

Therefore, the exponential of the original matrix B is

$\exp(B) = P \exp(J) P^{-1} = P \begin{bmatrix} e^4 & 0 & 0 \\ 0 & e^{16} & e^{16} \\ 0 & 0 & e^{16} \end{bmatrix} P^{-1} = {1\over 4} \begin{bmatrix} 13e^{16} - e^4 & 13e^{16} - 5e^4 & 2e^{16} - 2e^4 \\ -9e^{16} + e^4 & -9e^{16} + 5e^4 & -2e^{16} + 2e^4 \\ 16e^{16} & 16e^{16} & 4e^{16} \end{bmatrix}.$

Applications

Linear differential equations

The matrix exponential has applications to systems of linear differential equations. In Mathematics, a linear differential equation is a Differential equation of the form Ly = f \ where the Differential Recall from earlier in this article that a differential equation of the form

$\mathbf{y}' = C\mathbf{y}$

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

$\mathbf{y}(t) = \begin{pmatrix} y_1(t) \\ \vdots \\y_n(t) \end{pmatrix}$

we can express a system of coupled linear differential equations as

$\mathbf{y}'(t) = A\mathbf{y}(t)+\mathbf{b}(t).$

If we make an ansatz and use an integrating factor of e^−At and multiply throughout, we obtain

$e^{-At}\mathbf{y}'-e^{-At}A\mathbf{y} = e^{-At}\mathbf{b}$

$\frac{d}{dt} (e^{-At}\mathbf{y}) = e^{-At}\mathbf{b}.$

If we can calculate e^At, then we can obtain the solution to the system. In physics and mathematics an ansatz ( Ger, "anset onset" today "setup" plural Ansätze) is an educated guess that is

Example (homogeneous)

Say we have the system

$\begin{matrix} x' &=& 2x&-y&+z \\ y' &=& &3y&-1z \\ z' &=& 2x&+y&+3z \end{matrix}$

We have the associated matrix

$M=\begin{bmatrix} 2 & -1 & 1 \\ 0 & 3 & -1 \\ 2 & 1 & 3 \end{bmatrix}$

In the example above, we have calculated the matrix exponential

$e^{tM}=\begin{bmatrix} 2e^t - 2te^{2t} & -2te^{2t} & 0 \\ -2e^t + 2(t+1)e^{2t} & 2(t+1)e^{2t} & 0 \\ 2te^{2t} & 2te^{2t} & 2e^t\end{bmatrix}$

so the general solution of the system is

$\begin{bmatrix}x \\y \\ z\end{bmatrix}= C_1\begin{bmatrix}2e^t - 2te^{2t} \\-2e^t + 2(t+1)e^{2t}\\2te^{2t}\end{bmatrix} +C_2\begin{bmatrix}-2te^{2t}\\2(t+1)e^{2t}\\2te^{2t}\end{bmatrix} +C_3\begin{bmatrix}0\\0\\2e^t\end{bmatrix}$

that is,

$\begin{align} x & = C_1(2e^t - 2te^{2t}) + C_2(-2te^{2t}) \\ y & = C_1(-2e^t + 2(t+1)e^{2t})+C_2(2(t+1)e^{2t}) \\ z & = (C_1+C_2)(2te^{2t})+2C_3e^t \end{align}$

Inhomogeneous case - variation of parameters

For the inhomogeneous case, we can use integrating factors (a method akin to variation of parameters). In Mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given Ordinary differential equation. In Mathematics, variation of parameters also known as variation of constants, is a general method to solve inhomogeneous linear Ordinary We seek a particular solution of the form y_p(t) = exp(tA)z(t) :

$\mathbf{y}_p' = (e^{tA})'\mathbf{z}(t)+e^{tA}\mathbf{z}'(t)$

$= Ae^{tA}\mathbf{z}(t)+e^{tA}\mathbf{z}'(t)$

$= A\mathbf{y}_p(t)+e^{tA}\mathbf{z}'(t)$

For y_p to be a solution:

$e^{tA}\mathbf{z}'(t) = \mathbf{b}(t)$

$\mathbf{z}'(t) = (e^{tA})^{-1}\mathbf{b}(t)$

$\mathbf{z}(t) = \int_0^t e^{-uA}\mathbf{b}(u)\,du+\mathbf{c}$

So,

$\begin{align} \mathbf{y}_p & {} = e^{tA}\int_0^t e^{-uA}\mathbf{b}(u)\,du+e^{tA}\mathbf{c} \\ & {} = \int_0^t e^{(t-u)A}\mathbf{b}(u)\,du+e^{tA}\mathbf{c} \end{align}$

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

Example (inhomogeneous)

Say we have the system

$\begin{matrix} x' &=& 2x&-y&+z&+e^{2t} \\ y' &=& &3y&-1z& \\ z' &=& 2x&+y&+3z&+e^{2t} \end{matrix}$

So we then have

$M=\begin{bmatrix} 2 & -1 & 1 \\ 0 & 3 & -1 \\ 2 & 1 & 3 \end{bmatrix}$

and

$\mathbf{b}=e^{2t}\begin{bmatrix}1 \\0\\1\end{bmatrix}.$

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:

$\mathbf{y}_p = e^{t}\int_0^t e^{(-u)A}\begin{bmatrix}e^{2u} \\0\\e^{2u}\end{bmatrix}\,du+e^{tA}\mathbf{c}$

$\mathbf{y}_p = e^{t}\int_0^t \begin{bmatrix} 2e^u - 2ue^{2u} & -2ue^{2u} & 0 \\ \\ -2e^u + 2(u+1)e^{2u} & 2(u+1)e^{2u} & 0 \\ \\ 2ue^{2u} & 2ue^{2u} & 2e^u\end{bmatrix}\begin{bmatrix}e^{2u} \\0\\e^{2u}\end{bmatrix}\,du+e^{tA}\mathbf{c}$

$\mathbf{y}_p = e^{t}\int_0^t \begin{bmatrix} e^{2u}( 2e^u - 2ue^{2u}) \\ \\ e^{2u}(-2e^u + 2(1 + u)e^{2u}) \\ \\ 2e^{3u} + 2ue^{4u}\end{bmatrix}+e^{tA}\mathbf{c}$

$\mathbf{y}_p = e^{t}\begin{bmatrix} -{1 \over 24}e^{3t}(3e^t(4t-1)-16) \\ \\ {1 \over 24}e^{3t}(3e^t(4t+4)-16) \\ \\ {1 \over 24}e^{3t}(3e^t(4t-1)-16)\end{bmatrix}+ \begin{bmatrix} 2e^t - 2te^{2t} & -2te^{2t} & 0 \\ \\ -2e^t + 2(t+1)e^{2t} & 2(t+1)e^{2t} & 0 \\ \\ 2te^{2t} & 2te^{2t} & 2e^t\end{bmatrix}\begin{bmatrix}c_1 \\c_2 \\c_3\end{bmatrix}$

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

References

Roger A. In Mathematics, the logarithm of a matrix is a Matrix function which generalizes the scalar Logarithm to matrices. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In differential geometry the exponential map is a generalization of the ordinary Exponential function of mathematical analysis to all differentiable manifolds with an Affine In Mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a Vector field. A phase-type distribution is a Probability distribution that results from a system of one or more inter-related Poisson processes occurring in Sequence 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. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Module for the Matrix Exponential

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Contents

Properties

Linear differential equations

The exponential of sums

The exponential map

Computing the matrix exponential

Diagonalizable case

Nilpotent case

Generalization

Calculations

Applications

Linear differential equations

Example (homogeneous)

Inhomogeneous case - variation of parameters

Example (inhomogeneous)

See also

References

External links