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Concept
Matrix exponential
Summary
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 X be an n×n real or complex matrix. The exponential of X, denoted by eX or exp(X), is the n×n matrix given by the power series
where is defined to be the identity matrix with the same dimensions as . The series always converges, so the exponential of X is well-defined.
Equivalently,
where I is the n×n identity matrix.
If X is a 1×1 matrix the matrix exponential of X is a 1×1 matrix whose single element is the ordinary exponential of the single element of X.
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. The matrix exponential satisfies the following properties.
We begin with the properties that are immediate consequences of the definition as a power series:
e0 = I
exp(XT) = (exp X)T, where XT denotes the transpose of X.
exp(X∗) = (exp X)∗, where X∗ denotes the conjugate transpose of X.
If Y is invertible then eYXY−1 = YeXY−1.
The next key result is this one:
If then .
The proof of this 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. It is important to note that this identity typically does not hold if and do not commute (see Golden-Thompson inequality below).
Consequences of the preceding identity are the following:
eaXebX = e(a + b)X
eXe−X = I
Using the above results, we can easily verify the following claims. If X is symmetric then eX is also symmetric, and if X is skew-symmetric then eX is orthogonal. If X is Hermitian then eX is also Hermitian, and if X is skew-Hermitian then eX is unitary.
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