Expy

expY is a mathematical notation that can refer to the exponential operation applied to an object Y. In the scalar case, expY denotes the standard exponential function e^Y, defined for real (and complex) values. It is strictly increasing on the real line and satisfies basic laws such as exp(0) = 1 and exp(a + b) = exp(a)exp(b) when a and b commute.

When Y is a square matrix, exp(Y) usually means the matrix exponential, defined by the convergent power series exp(Y) = sum_{k=0}^∞ Y^k / k!. This matrix exponential is invertible, with inverse exp(-Y), and satisfies det(exp(Y)) = exp(tr(Y)). It plays a central role in solving linear differential equations and in studying linear dynamical systems. For diagonalizable matrices, exp(Y) can be computed as P exp(D) P^{-1}, where Y = P D P^{-1} and D is diagonal.

For a vector-valued argument, exp(Y) is often interpreted elementwise in some contexts (Hadamard exponentials), or it may be left undefined without additional structure. In many applications, especially in linear algebra and control theory, the matrix exponential is used to describe state evolution via y'(t) = Y y(t), with solution y(t) = exp(tY) y(0).

Computationally, the matrix exponential is commonly obtained through scaling and squaring combined with Padé approximants, or via eigenvalue decompositions when available.

Applications span differential equations, quantum mechanics, computer graphics, and stochastic processes, where exp(Y) provides a compact description of continuous-time evolution and growth. The concept has historical roots in the scalar exponential function and was extended to matrices by early 20th-century mathematicians, later becoming a fundamental tool in applied mathematics.