matrix calculus

matrix calculus - D . Matrix Calculus D1 D2 Appendix D:...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
. D Matrix Calculus D–1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Appendix D: MATRIX CALCULUS D–2 In this Appendix we collect some useful formulas of matrix calculus that often appear in finite element derivations. § D.1 THE DERIVATIVES OF VECTOR FUNCTIONS Let x and y be vectors of orders n and m respectively: x = x 1 x 2 . . . x n , y = y 1 y 2 . . . y m ,( D . 1 ) where each component y i may be a function of all the x j , a fact represented by saying that y is a function of x ,or y = y ( x ). ( D . 2 ) If n = 1, x reduces to a scalar, which we call x .I f m = 1, y reduces to a scalar, which we call y . Various applications are studied in the following subsections. § D.1.1 Derivative of Vector with Respect to Vector The derivative of the vector y with respect to vector x is the n × m matrix y x def = y 1 x 1 y 2 x 1 ··· y m x 1 y 1 x 2 y 2 x 2 y m x 2 . . . . . . . . . . . . y 1 x n y 2 x n y m x n ( D . 3 ) § D.1.2 Derivative of a Scalar with Respect to Vector If y is a scalar, y x def = y x 1 y x 2 . . . y x n .( D . 4 ) § D.1.3 Derivative of Vector with Respect to Scalar If x is a scalar, y x def = £ y 1 x y 2 x ... y m x ¤ ( D . 5 ) D–2
Background image of page 2
D–3 § D.1 THE DERIVATIVES OF VECTOR FUNCTIONS REMARK D.1 Many authors, notably in statistics and economics, define the derivatives as the transposes of those given above. 1 This has the advantage of better agreement of matrix products with composition schemes such as the chain rule. Evidently the notation is not yet stable. EXAMPLE D.1 Given y = ± y 1 y 2 i , x = " x 1 x 2 x 3 ² ( D . 6 ) and y 1 = x 2 1 x 2 y 2 = x 2 3 + 3 x 2 ( D . 7 ) the partial derivative matrix y /∂ x is computed as follows: y x = y 1 x 1 y 2 x 1 y 1 x 2 y 2 x 2 y 1 x 3 y 2 x 3 = " 2 x 1 0 13 02 x 3 ² ( D . 8 ) § D.1.4 Jacobian of a Variable Transformation In multivariate analysis, if x and y are of the same order, the determinant of the square matrix x /∂ y , that is J = ¯ ¯ ¯ ¯ x y ¯ ¯ ¯ ¯ ( D . 9 ) is called the Jacobian of the transformation determined by y = y ( x ) . The inverse determinant is J 1 = ¯ ¯ ¯ ¯ y x ¯ ¯ ¯
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/21/2012 for the course STAT 3008 taught by Professor C.y.yau during the Spring '11 term at CUHK.

Page1 / 8

matrix calculus - D . Matrix Calculus D1 D2 Appendix D:...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online