Morphometric Data - An Invariant Approach to Statistical Analysis of Shapes - page 50

For example, is a 3 by 3 identity matrix. In this

monograph we will denote such a matrix by I 3 which tells us
that the matrix is an identity matrix of a specific dimension,
in this case 3. Thus I D is a D by D identity matrix. An impor-
tant property of an identity matrix is that if we multiply any
matrix by an identity matrix, the resultant matrix is the same
as the original matrix.

b) An orthogonal matrix : A square matrix is an orthogonal

matrix if the product of itself and its transpose is an identity
matrix. In other words, a matrix R is an orthogonal matrix, if
RR T I .

Exercise : Verify that a matrix is an orthog-

R T R

onal matrix. To verify this result, recall that sin 2 ( - ) 1.

Because this matrix features in the rotation of an object by an
angle - , we denote it by R ( - ) . Thus, we write

cos 2 ( - )

R - =

.

We refer to R ( - ) also as the rotation matrix or the rotation
parameter.

c) Rotation of an object : Let the landmark coordinate matrix for

a given two-dimensional object be denoted by M . Suppose we
rotate this object by an angle - . The landmark coordinate
matrix corresponding to the rotated object, M , can be obtained
by multiplying the original landmark coordinate matrix by
the orthogonal matrix R ( - ) . Thus, we get M

MR ( - ).
Similarly, for a three dimensional object one can obtain the
landmark coordinates of the rotated object by multiplying the
original landmark coordinate matrix by a 3 by 3 orthogonal
matrix. A 3 by 3 orthogonal matrix is given by:

.