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单词 DerivativeOfMatrix
释义

derivative of matrix


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

Properties

In the below we assume that all matrices are dependent on a parameter tand 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

    ddt(AB)=dAdtB+AdBdt.
  3. 3.

    When A(t) is invertiblePlanetmathPlanetmath,

    ddt(A-1)=-A-1dAdtA-1.
  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 theHadamard product of A and B, then

    ddt(AB)=dAdtB+AdBdt.
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更新时间:2025/5/4 18:15:46