Matrixexponential

In mathematics, the Matrixexponential is a function of the amount of the square matrix, which is defined in analogy to the real exponential function. The Matrixexponential establishes the connection between Lie algebra and the corresponding Lie group.

• 5.1 diagonalizable matrices
• 5.2 nilpotent case
• 5.3 General case
• 5.4 Example
• 5.5 Numerical Methods

• 6.1 Linear Differential Equations 6.1.1 Example ( homogeneous)
• 6.1.2 Inhomogeneous case - variation of constants
• 6.1.3 Example ( inhomogeneous )

Definition

Be a real or complex matrix. The exponential of what is referred to by, or the matrix, which is defined by the following power series.

This series always converges. Therefore, the exponential is well defined. When a matrix is equal to that of the conventional exponential Matrixexponential. A generalization, which is also useful for infinite matrices, is the exponential function on arbitrary Banach algebras.

Properties

The Matrixexponential shares a number of characteristics of the ordinary exponential function. For example, the exponential of the zero matrix is equal to the unit matrix:

For arbitrary complex matrices and arbitrary complex numbers and is

It follows

That is

It refers to the inverse matrix.

The exponential function satisfies for all numbers and. The same is true for commuting matrices and, that is, from

Follows

For nichtkommutierende matrices, this equation is not true in general. In this case, one can calculate with the help of the Baker -Campbell - Hausdorff formula.

The exponential of the transposed matrix is equal to the transposition of the Exponentials of:

It follows that the matrix exponential maps symmetric matrices on symmetric matrices and skew-symmetric matrices are orthogonal matrices. Analogously, between adjunction and transposition, the relationship

So that the matrix exponential Hermitian matrices to Hermitian matrices and matrices schiefhermitesche maps to unitary matrices.

Furthermore:

• If is invertible, then.
• , Where denotes the trace of the square matrix.
• .

The exponential map

The exponential of a matrix is ​​always an invertible matrix. The inverse of is given by. Thus, the (complex) Matrixexponential provides image

From the vector space of all (complex) matrices in the general linear group, the group of all (complex) invertible matrices. This map is surjective, ie every ( real or complex ) invertible matrix can be used as the matrix exponential of a complex (!) Matrix can be written. The matrix logarithm returns the inverse of this figure.

For any two matrices and applies

Being an arbitrary matrix norm respectively. It follows that the exponential map is Lipschitz continuous and on compact subsets of even. For the norm of Matrixexponentials itself but there is a more precise bound

With the logarithmic matrix norm and the numerical range of values.

The assignment

Defines a smooth curve in the general linear group, which provides for the identity matrix. This provides a single parameter sub- group of the linear array as

Applies. The derivative of this function at the point is by

Given. The derivative of the matrix is even, that is, generates this single parameter subset.

More generally,

Examples of Lie algebras and Lie groups associated

In the last example can be seen that the exponential for the production of Lie groups (depending on the Lie algebra ) is generally not onto.

Linear Differential Equations

One of the advantages of Matrixexponentials is that you can use it to solve systems of linear ordinary differential equations. From equation (1 ) below follows, for example, that the solution of the initial value problem

Wherein a matrix is, by

Is given. The Matrixexponential can also help to solve the inhomogeneous equation

Be used. Examples can be found below in the section applications.

For differential equations of the form

With non- constant, there are no closed solutions. However, the Magnus series provides a solution as an infinite sum.

Calculation of Matrixexponentials

Diagonalizable matrices

The matrix is ​​a diagonal matrix

Then you can determine its exponential by applying the usual exponential function on each entry of the main diagonal:

Thus one can also calculate the exponential diagonalisierbarer matrices. if

Is a diagonal matrix and a change of basis matrix, then:

Nilpotent case

A matrix is ​​nilpotent if for a suitable natural number. In this case, the Matrixexponential can be calculated directly from the series development as the row terminates after a finite number of terms:

General case

Decomposes the minimal polynomial (or the characteristic polynomial ) of the matrix into linear factors (over which is always the case), then you can clearly into a sum

Be decomposed, with

• Is diagonalizable,
• Is nilpotent and
• Commutes with (that ).

Thus one can calculate the exponential of, by reducing it to the aforementioned cases. In the last step you need the commutativity of and.

Another ( closely related ) method is to use the Jordan normal form of. Be the Jordan normal form of the transition matrix, that is, it is

Because of

Applies

Therefore, one only needs the exponential of a Jordan block know. Now each Jordan block is of the form

With a special nilpotent matrix. The exponential of the Jordan block is therefore

Example

Consider the matrix

Which the Jordan form

With the transition matrix

Has. Then we have

And

Is therefore

The exponential of a 1 × 1 matrix is trivial. with the following

The Jordan normal form, and then calculate the exponential is very tedious, in this way. Usually, it is sufficient to calculate the effect of the exponential matrix of a few vectors.

Numerical Methods

The Jordan normal form decomposition is numerically unstable as a result of the floating point rounding errors are introduced in the eigenvalues ​​, which makes a grouping of the eigenvalues ​​impossible in groups of identical eigenvalues. Therefore, in the numerics other techniques to calculate the Matrixexponentials be used. One of the most effective available algorithms is the Padé approximation with scaling and squaring. Of large matrices, the computing effort can be further reduced by Krylowräume are used, the basis vectors are orthogonalized with the Arnoldi method.

Applications

Linear Differential Equations

The Matrixexponential can be used for the solution of a system of differential equations. A differential equation of the form

Has the solution. If the vector

Considered, then you can consider a system of coupled linear differential equations as

If one sets the integration factor and multiplied on both sides, one obtains

When calculated, to obtain a solution of the differential equation system.

Example ( homogeneous)

Consider the following system of differential equations

It can be written as with the coefficient matrix

Thus, the associated Matrixexponential results to

As a general solution of the differential equation system is obtained thereby

The inhomogeneous case - variation of constants

For the inhomogeneous case, one can use a similar method of variation of constants. It searches a solution of the form:

To find the solution that is given to

This results in

Being determined by the initial conditions.

Example ( inhomogeneous )

Consider the system of differential equations

With the matrix from above, the system writes

With

The general solution of the homogeneous equation was calculated above. The sum of the homogeneous and particular solutions yields the solution to the inhomogeneous problem. You have now found only a special solution ( via the variation of constants). From the equation above we get:

So