2.4 Derivatives
For later sections of this book, it will be useful
to introduce matrix notation for derivatives
of a scalar function of a vector
with respect to
.
Consider
and a
vector
, then
is the column vector
of partial derivatives
and
is the row vector of the same
derivative
(
is called the gradient
of
).
We can also introduce second order derivatives:
is the
matrix of elements
and
.
(
is
called the Hessian of
).
Suppose that
is a
vector and that
is a
matrix. Then
The Hessian of the quadratic form
is:
 |
(2.25) |
EXAMPLE 2.8
Consider the matrix
From formulas (
2.24) and (
2.25) it immediately
follows that the gradient of

is
and the Hessian is