Analytic function of a matrix - Wikipedia
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.
Matrix function | Definition, intuition, applications - Statlect
Learn how matrix functions are defined. Read an intuitive explanation of their definition. Discover how they are used.
Matrix function - Simple English Wikipedia, the free encyclopedia
In mathematics, a function maps an input value to an output value. In the case of a matrix function, the input and the output values are matrices . One example of a matrix function occurs with the Algebraic Riccati equation , which is used to solve certain optimal control problems.
These expressions implicitly apply as well to scalar-, vector-, or matrix-valued functions of scalar, vector, or matrix arguments. D.1.2.0.1 Example. Cubix. Suppose f(X) : R2×2→R2 = XTa and g(X) : R2×2→R2 = Xb. We wish to find ∇X ¡ f(X)Tg(X) ¢ = ∇X aTX2b (2068) using the product rule. Formula (2066) calls for ∇X a TX2b = ∇ X(X
functions (we will see later when we investigate more the connection between functions and matrices that this is the correct rule). At this point a few simple examples will probably help.
Matrix Function: Simple Definition, Examples - Statistics How To
A matrix function can be defined in many ways with real or complex numbers. It usually involves one square matrix mapping to another matrix. Examples, more definitions.
Functions of Matrices | SIAM Publications Library
Theory of Matrix Functions. 2. Applications. 3. Conditioning. 4. Techniques for General Functions. 5. Matrix Sign Function. 6. Matrix Square Root. 7. Matrix p th Root. 8. The Polar …
§D.3.1 Functions of a Matrix Determinant An important family of derivatives with respect to a matrix involves functions of the determinant of a matrix, for example y =|X| or y =|AX|. Suppose that we have a matrix Y = [yij] whose components are functions of a matrix X = [xrs], that is yij = fij(xrs), and set out to build the matrix ∂|Y| ∂X ...
Functions of matrices 1.1 2×2 matrices with distinct eigenvalues For each matrix below, the second form can be used to compute functions of matrices, such as the matrix exponential. For example, if A = 1 2 3 2 = −1 2 1 3 −1 0 0 4 −3 2 1 1 /5 then A = −1 3 −2 −3 3 /5 + 4 2 2 3 3 An = (−1)n 3 −2 −3 3 /5 + 4n 2 2 3 3 eAt = e−t ...
Functions of Matrices | 1. Theory of Matrix Functions
2012年3月23日 · • Functions mapping ℂ n×n to ℂ m×m that do not stem from a scalar function. Examples include matrix polynomials with matrix coefficients, the matrix transpose, the adjugate (or adjoint) matrix, compound matrices comprising minors of a given matrix, and factors from matrix factorizations.
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