Matrix Calculus
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Date Submitted: 10/06/2012 08:25 AM
.
D
Matrix
Calculus
D–1
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 =
x1
x2
...
xn
, y =
y1
y2
...
ym
, (D.1)
where each component yi may be a function of all the xj , 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
=
∂y1
∂x1
∂y2
∂x1 · · · ∂ym
∂x1
∂y1
∂x2
∂y2
∂x2 · · · ∂ym
∂x2
...
...
. . .
...
∂y1
∂xn
∂y2
∂xn · · · ∂ym
∂xn
(D.3)
§D.1.2 Derivative of a Scalar with Respect to Vector
If y is a scalar,
∂y
∂x
def
=
∂y
∂x1
∂y
∂x2
...
∂y
∂xn
. (D.4)
§D.1.3 Derivative of Vector with Respect to Scalar
If x is a scalar,
∂y
∂x
def
=
∂y1
∂x
∂y2
∂x . . .
∂ym
∂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 =
y1
y2
, x =
x1
x2
x3
(D.6)
and
y1 = x2
1 − x2
y2 = x2
3 + 3x2
(D.7)
the partial derivative matrix ∂y/∂x is computed as follows:
∂y
∂x =
∂y1
∂x1
∂y2
∂x1
∂y1
∂x2
∂y2
∂x2
∂y1
∂x3
∂y2
∂x3
=
2x1 0
−1 3
0 2x3
(D.8)
§D.1.4 Jacobian of a Variable Transformation
In multivariate analysis, if x and y are of the same order,...
