Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 406

APPENDIX

B

Background Information
and Techniques

B.1 Vectors and matrices
A vector is a one-dimensional list of values. This list can be shown as a row vector or a column vector
(e.g., Eq. B.1 ) . In general, a matrix is an n -dimensional array of values. For purposes of this topic, a
matrix is two-dimensional (e.g., Eq. B.2 ) .

2 3

a
b
c

4 5

½abc (B.1)

2 3

abc
def
ghi

4 5

(B.2)

Matrices are multiplied together by taking the i th row of the first matrix and multiplying each
element by the corresponding element of the j th column of the second matrix and summing all

the products to produce the i , j th element. When computing C ¼ AB , where A has v elements

in each row and B has v elements in each column, an element C ij is computed according to

Equation B.3 .

C ij ¼ A i 1 B 1 j þ A i 2 B 2 j þ A i 3 B 3 j þ ...þ A iv B vj

X v
k¼ 1 A ik B kj

(B.3)

¼

The “inside” dimension of the matrices must match in order for the matrices to be multiplied
together. That is, if A and B are multiplied and A is a matrix with U rows and V columns (a U V
matrix), then B must be a V W matrix; the result will be a U W matrix. In other words, the number
of columns (the number of elements in a row) of A must be equal to the number of rows (the number of
elements in a column) of B. As a more concrete example, consider multiplying two 3

3 matrices.

Equation B.4 shows the computation for the first element.