section_path: Integer powers in algebra/Matrices and linear operators
section_name: Matrices and linear operators
paragraph_text: Matrices and linear operators
  
  If A is a square matrix, then the product of A with itself n times is called the matrix power. Also A 0 {\displaystyle A^{0}} is defined to be the identity matrix, and if A is invertible, then A − n = (A − 1) n {\displaystyle A^{-n}=\left(A^{-1}\right)^{n}}.
  
  Matrix powers appear often in the context of discrete dynamical systems, where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system. This is the standard interpretation of a Markov chain, for example. Then A 2 x {\displaystyle A^{2}x} is the state of the system after two time steps, and so forth: A n x {\displaystyle A^{n}x} is the state of the system after n time steps. The matrix power A n {\displaystyle A^{n}} is the transition matrix between the state now and the state at a time n steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors.
  
  Apart from matrices, more general linear operators can also be exponentiated. An example is the derivative operator of calculus, d / d x {\displaystyle d/dx}, which is a linear operator acting on functions f (x) {\displaystyle f(x)} to give a new function (d / d x) f (x) = f ′ (x) {\displaystyle (d/dx)f(x)=f'(x)}. The n th power of the differentiation operator is the n th derivative:
  
  These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups. Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.