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Wikipedia / Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group.

Let be an real or complex matrix. The exponential of , denoted by , , or , is the matrix given by the power series

, and for integer . The series always converges, so the exponential of is well-defined.

for integer-valued , where is the identity matrix.

of the (matrix) differential equation

evaluated at .

When is an diagonal matrix then will be an diagonal matrix with each diagonal element equal to the ordinary exponential function applied to the corresponding diagonal element of .

Properties

Elementary properties

Let and be complex matrices and let and be arbitrary complex numbers. We denote the identity matrix by and the zero matrix by 0. The matrix exponential satisfies the following properties.

We begin with the properties that are immediate consequences of the definition as a power series:

The proof of this last identity is the same as the standard power-series argument for the corresponding identity for the exponential of real numbers. That is to say, commute do not commute (see Golden-Thompson inequality below). Special cases of this identity include:

Using the above results, we can easily verify the following claims:

Finally, a Laplace transform of matrix exponentials amounts to the resolvent for all sufficiently large positive values of .

Linear differential equation systems

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 where is a constant matrix and y

See the section on applications below for examples.

where is not constant, but the Magnus series gives the solution as an infinite sum.

The determinant of the matrix exponential

By Jacobi's formula, for any complex square matrix the following trace identity holds:

In addition to providing a computational tool, this formula demonstrates that a matrix exponential is always an invertible matrix. This follows from the fact that the right hand side of the above equation is always non-zero, and so , which implies that must be invertible.

to not be surjective, in contrast to the complex case mentioned earlier. This follows from the fact that, for real-valued matrices, the right-hand side of the formula is always positive, while there exist invertible matrices with a negative determinant.

Real symmetric matrices

be an a column vector. Using the elementary properties of the matrix exponential and of symmetric matrices, we have:

is positive definite.

Tensor product of exponential

The exponential of the Kronecker sum which must not be confused with the direct sum In this case the exponential is simply the tensor product of the exponentials of the matrices:

  1. is the Identity matrix.

  2. This follows from the commutation of the summands of the Kronecker sum and the properties discussed above.

This result is tied to the Direct Product of Lie groups as a representation of the Direct sum of Lie algebras.

leads to the additivity of the von Neumann entropy for independent systems if these logarithmic expressions exist.

The exponential of sums

For any numbers (scalars) and we know that the exponential function satisfies . The same is true for commuting matrices. If matrices and commute (meaning that

However, for matrices that do not commute the above equality does not necessarily hold.

The Lie product formula

Even if and do not commute, the exponential can be computed by the Lie product formula

Using a large finite to approximate the above is basis of the Suzuki-Trotter expansion, often used in numerical time evolution.

The Baker–Campbell–Hausdorff formula

In the other direction, if and where may be computed as a series in commutators of and by means of the Baker–Campbell–Hausdorff formula: where the remaining terms are all iterated commutators involving and . If and commute, then all the commutators are zero and we have simply .

Inequalities for exponentials of Hermitian matrices

For Hermitian matrices there is a notable theorem related to the trace of matrix exponentials.

If and are Hermitian matrices, then

There is no requirement of commutativity. There are counterexamples to show that the Golden–Thompson inequality cannot be extended to three matrices – and, in any event, is not guaranteed to be real for Hermitian , , . However, Lieb proved

The exponential map

The exponential of a matrix is always an invertible matrix. The inverse matrix of is given by from the space of all n × n matrices to the general linear group of degree , i.e. the group of all n × n invertible matrices. In fact, this map is surjective which means that every invertible 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).

For any two matrices and

where denotes an arbitrary matrix norm. It follows that the exponential map is continuous and Lipschitz continuous on compact subsets of .

defines a smooth curve in the general linear group which passes through the identity element at .

In fact, this gives a one-parameter subgroup

The derivative of this curve (or tangent vector) at a point t is given by The derivative at is just the matrix X, which is to say that X generates this one-parameter subgroup.

More generally, for a generic -dependent exponent, ,

Taking the above expression outside the integral sign and expanding the integrand with the help of the Hadamard lemma one can obtain the following useful expression for the derivative of the matrix exponent,

The coefficients in the expression above are different from what appears in the exponential. For a closed form, see derivative of the exponential map.

Directional derivatives when restricted to Hermitian matrices

, the directional derivative is is

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. Matlab, GNU Octave, R, and SciPy all use the Padé approximant. In this section, we discuss methods that are applicable in principle to any matrix, and which can be carried out explicitly for small matrices. Subsequent sections describe methods suitable for numerical evaluation on large matrices.

Diagonalizable case

If a matrix is diagonal

This result also allows one to exponentiate diagonalizable matrices. If then which is especially easy to compute when is diagonal.

Application of Sylvester's formula yields the same result. (To see this, note that addition and multiplication, hence also exponentiation, of diagonal matrices is equivalent to element-wise addition and multiplication, and hence exponentiation; in particular, the "one-dimensional" exponentiation is felt element-wise for the diagonal case.)

Example : Diagonalizable

Nilpotent case

A matrix is nilpotent if for some integer q. In this case, the matrix exponential can be computed directly from the series expansion, as the series terminates after a finite number of terms:

Since the series has a finite number of steps, it is a matrix polynomial, which can be computed efficiently.

General case

Using the Jordan–Chevalley decomposition

By the Jordan–Chevalley decomposition matrix X where

  • A is diagonalizable

  • N is nilpotent

  • A commutes with N

This means that we can compute the exponential of X

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

Using the Jordan canonical form

A closely related method is, if the field is algebraically closed, to work with the Jordan form of . Suppose that where is the Jordan form of

Therefore, we need only know how to compute the matrix exponential of a Jordan block

where is a special nilpotent matrix. The matrix exponential of

Projection case

If is a projection matrix (i.e. is idempotent: ), its matrix exponential is:

Deriving this by expansion of the exponential function, each power of reduces to

Rotation case

For a simple rotation in which the perpendicular unit vectors and specify a plane, the rotation matrix can be expressed in terms of a similar exponential function involving a generator and angle .

The formula for the exponential results from reducing the powers of in the series expansion and identifying the respective series coefficients of and with and respectively. The second expression here for is the same as the expression for in the article containing the derivation of the generator, .

reduces to the standard matrix for a plane rotation.

The matrix projects a vector onto the -plane and the rotation only affects this part of the vector. An example illustrating this is a rotation of in the plane spanned by and ,

Let , so and its products with and are zero. This will allow us to evaluate powers of .

Evaluation by Laurent series

By virtue of the Cayley–Hamilton theorem the matrix exponential is expressible as a polynomial of order −1.

If and are nonzero polynomials in one variable, such that , and if the meromorphic function is entire To prove this, multiply the first of the two above equalities by and replace by .

Such a polynomial can be found as follows−see Sylvester's formula. Letting be a root of , is solved from the product of by the principal part of the Laurent series of at : It is proportional to the relevant Frobenius covariant. Then the sum S of the Q, where runs over all the roots of , can be taken as a particular . All the other Q will be obtained by adding a multiple of to . In particular, , the Lagrange-Sylvester polynomial, is the only whose degree is less than that of .

Example: Consider the case of an arbitrary

The exponential matrix , by virtue of the Cayley–Hamilton theorem

(For any complex number and any C-algebra , we denote again by the product of by the unit of .)

Let and be the roots of the characteristic polynomial of

if ; while, if

where is 0 if , and if .

Thus,

Thus, as indicated above, the matrix

the matrix exponential reduces to a plain product of the exponentials of the two respective pieces. This is a formula often used in physics, as it amounts to the analog of Euler's formula for Pauli spin matrices, that is rotations of the doublet representation of the group SU(2).

The polynomial can also be given the following "interpolation" characterization. Define , and . Then is the unique degree polynomial which satisfies whenever is less than the multiplicity of as a root of . We assume, as we obviously can, that is the minimal polynomial of . We further assume that is a diagonalizable matrix. In particular, the roots of are simple, and the "interpolation" characterization indicates that is given by the Lagrange interpolation formula, so it is the Lagrange−Sylvester polynomial.

At the other extreme, if

with

Evaluation by implementation of Sylvester's formulaSylvester's formula

A practical, expedited computation of the above reduces to the following rapid steps. Recall from above that an matrix amounts to a linear combination of the first −1 powers of by the Cayley–Hamilton theorem. For diagonalizable matrices, as illustrated above, e.g. in the case, Sylvester's formula yields , where the s are the Frobenius covariants of .

It is easiest, however, to simply solve for these s directly, by evaluating this expression and its first derivative at , in terms of and , to find the same answer as above.

But this simple procedure also works for defective matrices, in a generalization due to Buchheim. This is illustrated here for a example of a matrix which is not diagonalizable, and the s are not projection matrices.

with eigenvalues and , each with a multiplicity of two.

Consider the exponential of each eigenvalue multiplied by , . Multiply each exponentiated eigenvalue by the corresponding undetermined coefficient matrix . If the eigenvalues have an algebraic multiplicity greater than 1, then repeat the process, but now multiplying by an extra factor of for each repetition, to ensure linear independence.

. By contrast, when all eigenvalues are distinct, the s are just the Frobenius covariants, and solving for them as below just amounts to the inversion of the Vandermonde matrix of these 4 eigenvalues.)

To solve for all of the unknown matrices in terms of the first three powers of and the identity, one needs four equations, the above one providing one such at = 0. Further, differentiate it with respect to

(In the general case, −1 derivatives need be taken.)

Setting = 0 in these four equations, the four coefficient matrices

Substituting with the value for

The procedure is much shorter than Putzer's algorithm sometimes utilized in such cases.

Illustrations

Its Jordan form where the matrix P

Let us first calculate exp(J

The exponential of a matrix is just the exponential of the one entry of the matrix, so . The exponential of J(16) can be calculated by the formula mentioned above; this yields

Therefore, the exponential of the original matrix

Applications

Linear differential equations

The matrix exponential has applications to systems of linear differential equations. (See also matrix differential equation.) Recall from earlier in this article that a homogeneous has solution .

we can express a system of inhomogeneous Making an ansatz to use an integrating factor of

The second step is possible due to the fact that, if , then . So, calculating leads to the solution to the system, by simply integrating the third step with respect to .

.

Example (homogeneous)

The associated defective matrix

Example (inhomogeneous)

From before, we already 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, we now only need find the particular solution.

which could be further simplified to get the requisite particular solution determined through variation of parameters. Note c = y(0). For more rigor, see the following generalization.

Inhomogeneous case generalization: variation of parameters

For the inhomogeneous case, we can use integrating factors (a method akin to variation of parameters). We seek a particular solution of the form

For

where is determined by the initial conditions of the problem.

with the initial condition , where

  • is an by complex matrix,

  • is a continuous function from some open interval to ,

  • is a point of , and

  • is a vector of .

Left-multiplying the above displayed equality by

for

where the notation is as follows:

  • is a monic polynomial of degree ,

  • is a continuous complex valued function defined on some open interval ,

  • is a point of ,

  • is a complex number, and

in Subsection Evaluation by Laurent series above.

To justify this claim, we transform our order scalar equation into an order one vector equation by the usual reduction to a first order system where is the transpose companion matrix of . We solve this equation as explained above, computing the matrix exponentials by the observation made in Subsection Evaluation by implementation of Sylvester's formula above.

In the case

where the functions and are as in Subsection Evaluation by Laurent series above.

See also

References

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