MatrixEponenential-指数矩阵计算

is invertible then .

symmetric, and that if X is skew-symmetric then e X is orthogonal.

exp(X*) = (e 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.

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

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.

The exponential of sums

We know that the exponential function satisfies e x + y = e x e 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.

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

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.

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).

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

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

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

Diagonalizable case

If a matrix is diagonal:

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 D U-1.

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:

General case

An arbitrary matrix X (over an algebraically closed field) can be expressed uniquely as sum

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

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

Consider the matrix

which has Jordan form

and transition matrix

Now,

and

So,

The exponential calculation for a 1×1 matrix is clearly trivial, with e J1(4)=e4 so,

Clearly, to calculate the Jordan form and to evaluate the exponential this way is very tedious. Often, it will often suffice to calculate the action of the exponential matrix upon some vector in applications, and there are other methods available to achieve this.

Applications

Linear differential equations

The matrix exponential has applications to systems of linear differential equations. Recall that a differential equation of the form

y' = C y

has solution e C x. 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

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) :

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: