derivative of matrix

Suppose I is an open set of , and for each tI, A(t) is an n×m matrix. If each element in A(t) is a differentiable function of t, we say that A is a differentiableMathworldPlanetmath, and define the derivative of A componentwise. This derivative we shall write as ddtA or dAdt.


In the below we assume that all matrices are dependent on a parameter t and the matrices are differentiable with respect to t.

  1. 1.

    For any n×m matrix A,

    (dAdt)T = ddt(AT),

    where T is the matrix transpose.

  2. 2.

    If A(t),B(t) are matrices such that AB is defined, then

  3. 3.

    When A(t) is invertiblePlanetmathPlanetmath,

  4. 4.

    For a square matrixMathworldPlanetmath A(t),

    tr(dAdt) = ddttr(A),

    where tr is the matrix trace.

  5. 5.

    If A(t),B(t) are n×m matrices and AB is the Hadamard product of A and B, then

Title derivative of matrix
Canonical name DerivativeOfMatrix
Date of creation 2013-03-22 15:00:28
Last modified on 2013-03-22 15:00:28
Owner matte (1858)
Last modified by matte (1858)
Numerical id 10
Author matte (1858)
Entry type Definition
Classification msc 15-01
Related topic NthDerivativeOfADeterminant