Page 124 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

112 Chapter 3

As mentioned earlier the any vector ‘u’ in the space 1 and the
corresponding mapped vector v in the space 2 can also have their associated
coefficient vectors. Let [p1, p2, p3, …pn] be the co-efficient vector associated
with the vector ‘u’ and [q1, q2, q3…qn] be the co-efficient vector associated
with the vector ‘v’.
The matrix relating the co-efficient vector of the particular vector in the
space 1 and the co-efficient vector of the corresponding mapped vector in
the space 2 is called transformation matrix and is computed as described
below.

(i.e.)

T
T
[p1 p2 p3 p4 …pn] [TM] = [q1 q2 q3 q4 … qn]

The transformation matrix is obtained using the co-efficient vectors
computed for the T(u1), T(u2),…T(un) in the space 2 where u1,u2 u3,..un
are the independent vectors which spans the space 1.

Thus TM = a11 a21 a31 a41 a51 …an1
a12 a22 a32 a42 a52 …an2
a13 a23 a33 a43 a53 …an3
a14 a24 a34 a44 a54 …an4
…
a1n a2n a3n a4n a5n …ann

For example the vector u1 with co-efficient vector [1 0 0 0 0 0 0…0] is
mapped to the vector T(u1) whose co-efficient vector is obtained using TM
as

a11 a21 a31 a41 a51 …an1 1 a11
a12 a22 a32 a42 a52 …an2 0 a12

a13 a23 a33 a43 a53 …an3 0 = a13
a14 a24 a34 a44 a54 …an4 0 a14
…
a1n a2n a3n a4n a5n …ann 0 a1n

The co-efficient vector obtained is as expected.