Matrix Exponential - Properties

Properties

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:

• e0 = I
• eaXebX = e(a + b)X
• eXe−X = I
• If XY = YX then eXeY = eYeX = e(X + Y).
• If Y is invertible then eYXY−1 =YeXY−1.
• exp(XT) = (exp X)T, where XT denotes the transpose of X. It follows that if X is symmetric then eX is also symmetric, and that if X is skew-symmetric then eX is orthogonal.
• exp(X*) = (exp X)*, where X* denotes the conjugate transpose of X. It follows that if X is Hermitian then eX is also Hermitian, and that if X is skew-Hermitian then eX is unitary.