Mathematical Methods for Physicists: A concise introduction - Site Map

THE GRAM±SCHMIDT ORTHOGONALIZATION PROCESS
We now expand it in a new orthonormal basis je 1 i; je 2 i with components
je 1 i ˆ 1
p 1
; je 2 i ˆ 1
1
p :
2 1
2 1
To do this, let us write
jUi ˆ u 1 je 1 i‡ u 2 je 2 i
and determine u 1 and u 2 . To determine u 1 , we take the inner product of both sides
with he 1 j:
likewise,
u 1 ˆ he 1
jUi ˆ p 1 … 1 1†
2
u 2 ˆ p 1 …1
2
!
1 ‡ i
p ˆ 1 p
p …1 ‡
3 ‡ 2i†;
3 ‡ i 2
p
3 †:
As a check on the calculation, let us compute the norm squared of the vector and
see if it equals j1 ‡ ij 2 p
‡j 3 ‡ ij2 ˆ 6. We ®nd
ju 1 j 2 ‡ u 2
j j 2 ˆ 1
2
…1 ‡ 3 ‡ 2
3
p p
‡ 4 ‡ 1 ‡ 3 2 3 †ˆ6:
The Gram±Schmidt orthogonalization process
We now take up the Gram±Schmidt orthogonalization method for converting a
linearly independent basis into an orthonormal one. The basic idea can be clearly
illustrated in the following steps. Let j1i; j2i; ...; jii; ...be a linearly independent
basis. To get an orthonormal basis out of these, we do the following:
Step 1.
Step 2.
Rescale the ®rst vector by its own length, so it becomes a unit vector.
This will be the ®rst basis vector.
je 1 i ˆ j1i
jj1ij ;
where j1i
p
j j ˆ h
1 j 1i. Clearly
he 1 j e 1 i ˆ h1
j 1i
ˆ 1:
jj1ij
To construct the second member of the orthonormal basis, we subtract
from the second vector j2i its projection along the ®rst, leaving behind
only the part perpendicular to the ®rst.
jIIi ˆ j2i je 1 ihe 1 j2i:
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