Matrix function

In mathematics, a matrix function is a function which maps a matrix to another matrix. In mathematics, a matrix function is a function which maps a matrix to another matrix. There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained. All of the following techniques yield the same matrix function, but the domains on which the function is defined may differ. If the real function f has the Taylor expansion then a matrix function can be defined by substituting x by a matrix: the powers become matrix powers, the additions become matrix sums and the multiplications become scaling operations. If the real series converges for | x | < r {displaystyle |x|<r} , then the corresponding matrix series will converge for matrix argument A if ‖ A ‖ < r {displaystyle |A|<r} for some matrix norm ‖ ⋅ ‖ {displaystyle |cdot |} which satisfies ‖ A B ‖ ≤ ‖ A ‖ ⋅ ‖ B ‖ {displaystyle |AB|leq |A|cdot |B|} . If the matrix A is diagonalizable, the problem may be reduced to an array of the function on each eigenvalue.This is to say we can find a matrix P and a diagonal matrix D such that A = P ⋅ D ⋅ P − 1 {displaystyle A=Pcdot Dcdot P^{-1}} .Applying the power series definition to this decomposition, we find that f(A) is defined by where d 1 , … , d n {displaystyle d_{1},dots ,d_{n}} denote the diagonal entries of D. For example, suppose one is seeking A! = Γ(A+1) for