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
where is defined to be the identity matrix with the same dimensions as , and for integer . Equation 2.1 The series always converges, so the exponential of is well-defined.
Equivalently,
for integer-valued , where is the identity matrix.
Equivalently, the matrix exponential is provided by the solution 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. Proposition 2.3
We begin with the properties that are immediate consequences of the definition as a power series:
, where denotes the transpose of . , where denotes the conjugate transpose of . If is invertible then If then .
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, as long as and commute, it makes no difference to the argument whether and are numbers or matrices. This identity typically does not hold if and 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:
If is symmetric then is also symmetric.
If is skew-symmetric then is orthogonal.
If is Hermitian then is also Hermitian.
If is skew-Hermitian then is unitary.
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. The solution of
where is a constant matrix and y is a column vector, 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 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: Theorem 2.12
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.
In the real-valued case, the formula also exhibits the map
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
The matrix exponential of a real symmetric matrix is positive definite. Let be an real symmetric matrix and a column vector. Using the elementary properties of the matrix exponential and of symmetric matrices, we have:
Since is invertible, the equality only holds for , and we have for all non-zero . Hence is positive definite.
Tensor product of exponential
The exponential of the Kronecker sumBrewer, J.W. (1969). "A note on Kronecker matrix products and matrix equation systems". SIAM Journal on Applied Mathematics. 17 (3): 603–606. doi:10.1137/0117057. of two square matrices which must not be confused with the direct sum takes a simple form.
In this case the exponential is simply the tensor product of the exponentials of the matricesWeisstein, Eric W. "Kronecker Sum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/KroneckerSum.html :
Here we assumed to be of order respectively and is the Identity matrix of order . 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 and its associated Lie algebra for which
as a representation of the Direct sum of Lie algebras.
Another application of this formula is the physics of non-interacting systems. Its reverse formula
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 ), then
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 Theorem 2.11
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 are sufficiently small (but not necessarily commuting) matrices, we have
where may be computed as a series in commutators of and by means of the Baker–Campbell–Hausdorff formula: Chapter 5
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 that it can be generalized to three matrices if we modify the expression as follows
The exponential map The exponential of a matrix is always an invertible matrix. The inverse matrix of is given 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
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 Exercises 2.9 and 2.10 (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 .
The map
defines a smooth curve in the general linear group which passes through the identity element at .
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 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, Theorem 5.4
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
Let be a Hermitian matrix with distinct eigenvalues. Let be its eigen-decomposition where is a unitary matrix whose columns are the eigenvectors of , is its conjugate transpose, and the vector of corresponding eigenvalues. Then, for any Hermitian matrix , the directional derivative of at in the direction is See Theorem 3.3. See Propositions 1 and 2.
where , the operator denotes the Hadamard product, and, for all , the matrix is defined as
In addition, for any Hermitian matrix , the second directional derivative in directions and is
where the matrix-valued function is defined, for all , as
with
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.See Section 2.2 Subsequent sections describe methods suitable for numerical evaluation on large matrices.
Diagonalizable case
If a matrix is diagonal:
then its exponential can be obtained by exponentiating each entry on the main 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 For example, the matrix
can be diagonalized as
Thus,
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, any matrix X with complex entries can be expressed as
where A is diagonalizable N is nilpotent A commutes with N
This means that 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.
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 . 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 is a special nilpotent matrix. The matrix exponential of is then given by
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 which becomes a common factor of the sum:
Rotation case For a simple rotation in which the perpendicular unit vectors and specify a plane,in a Euclidean space 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, .
In two dimensions, if and , then , , and
reduces to the standard matrix for a plane rotation.