{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

matrix calculus

# matrix calculus - D Matrix Calculus D1 D2 Appendix D MATRIX...

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

. D Matrix Calculus D–1

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

View Full Document
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 . If 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
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 , x = x 1 x 2 x 3 ( 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 1 3 0 2 x 3 ( 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 . ( D . 10 ) 1

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 8

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

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

View Full Document
Ask a homework question - tutors are online