2.8 Matrices And Calculus
The study of moving coordinate systems gives a non trivial example of the usefulness of the ideas
involving linear transformations and matrices. To begin with, here is the concept of the product
rule extended to matrix multiplication.
Definition 2.8.1 Let A
be an m × n matrix. Say A
. Suppose also that
is a differentiable function for all i,j. Then define A′
. That is, A′
is the matrix which consists of replacing each entry by its derivative. Such an m×n matrix
in which the entries are differentiable functions is called a differentiable matrix.
The next lemma is just a version of the product rule.
Lemma 2.8.2 Let A
be an m × n matrix and let B
be an n × p matrix with the property
that all the entries of these matrices are differentiable functions. Then
Proof: This is like the usual proof.
and now, using the fact that the entries of the matrices are all differentiable, one can pass to a
limit in both sides as h → 0 and conclude that