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3. Numerical Linear Algebra 117

11.3 Transformation Matrix for Obtaining Co-efficient
of Eigen Basis

Consider the group of two dimensional vectors collected randomly forms the
vector space. This is consider as the subspace spanned by the independent
T
T
vectors u1 = [1 0 ] and u2=[0 1] . The subspace is spanned by the Eigen
T
T
basis E1=[-0.2459 -0.9693] and E 2 =[-0.9693 0.2459] (see section 2-1).
The vector [1 0] is represented as the linear combination u1 and u2 as [1
0] = 1*u1+0*u2.The transformed vector of [1 0] is the same vectors itself
(i.e.)[1 0].The vector [1 0] is represented as the linear combinations of Eigen
basis given by -0.2459*E1+(-0.9693)*E2 =[ 1 0 ]. Similarly the vector
T
[0 1] is represented as the linear combination of u1 and u2 as [0 1] =
0*u1+1*u2 and its transformed vector [0 1] is represented as the linear
combination of E1 and E2 as given by -0.9693*E1 +0.2459*E2.
Thus the transformation matrix which converts the co-efficients of the
basis u1 and u2 into the co-efficients of the Eigen basis E1 and E2 for the
particular vector in the space.

TM = -0.2459 -0.9693

0.9693 -0.2459
-

Note that the transformation matrix consists of Eigen basis arranged
row wise.

11.4 Transformation Matrix for Obtaining Co-efficient
of Wavelet Basis

Consider the 8-dimensional space spanned by the 8 independent vectors

T
u1= [1 0 0 0 0 0 0 0] u2= [0 1 0 0 0 0 0 0] T u3= [0 0 1 0 0 0 0 0] T
T
u4= [0 0 0 1 0 0 0 0] u5= [0 0 0 0 1 0 0 0] T u6= [0 0 0 0 0 1 0 0] T
T
u7= [0 0 0 0 0 0 1 0] u8= [0 0 0 0 0 0 0 1] T