mathmatical physics

1. NPTEL – Physics – Mathematical Physics - 1
Lecture 12
Gram-Schmidt Orthogonalization (GSO) [continued]
Example 1.
Two vectors u and v where 𝑢, 𝑣 ∈ 𝑉 are said to be orthogonal if < 𝑢|𝑣 > = 0. Similarly a
real matrix P is said to be orthogonal if P is invertible and if 𝑃𝑇 = 𝑃−1 or
𝑃𝑃𝑇 = 𝑃𝑇𝑃 = 1. Besides the rows and columns of P form an orthonormal set of vectors.
Also det |𝑃| = ±1.
Suppose 𝑤 ≠ 0, let v be any vector in V, it needs to be shown that,
𝑐 = <𝑣|𝑤>
= <𝑣|𝑤>
<𝑤|𝑤> ‖𝑤‖2
is the unique scalar such that
𝑉 = 𝑣 − 𝑐𝑤 is orthogonal to w.
The proof can proceed as follows. In order for 𝑉 to be orthogonal to w,
we must have
< 𝑣 − 𝑐𝑤|𝑤 > = 0 or, < 𝑣|𝑤 > −𝑐 < 𝑤|𝑤 ≥ 0.
or, < 𝑣|𝑤 > = 𝑐 < 𝑤|𝑤 >
or, 𝑐 = <𝑣|𝑤>
<𝑤|𝑤>
Then, < (𝑣 − 𝑐𝑤)|𝑤 > = < 𝑣|𝑤 > − 𝑐 < 𝑤|𝑤 >
= < 𝑣 |𝑤 > − <𝑣|𝑤>
< 𝑤|𝑤 > = 0
<𝑤|𝑤>
The above scalar c is called the fourier coefficient of v with respect to w or the
component of v along w. cw is the projection of v along w as shown below
Example 2.
Find the coefficient c and the projection cw of
𝑣 = (1, −1, 2)
𝑤 = (0, 1, 1) in 𝑅3.
Solution: < 𝑣|𝑤 > = 1 and ‖𝑤‖2 = 2
along
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2. NPTEL – Physics – Mathematical Physics - 1
𝑐 =
< 𝑣|𝑤 > 1
=
2
‖𝑤‖2
𝑐𝑤 = (0, 1
, 1
) is the projection of v along w.
2 2
Example 3.
Consider the following basis of Euclidean
space
𝑅3: {𝑣1 = (1, 1, 1), 𝑣2 = (0, 1, 1), 𝑣3 = (0, 0, 1)}. Use the Gram-Schmidt procedure to
transform {𝑣𝑖} into an orthonormal basis {𝑢𝑖} in 𝑅3.
Solution: To begin with, set 𝑤1 = 𝑣1 = (1, 1, 1)
Then find 𝑣2 −
<𝑣2|𝑤1>
‖𝑤 ‖2
1
𝑤1 = (0, 1, 1) − 3
(1, 1, 1)
2
= (− 2
, 1
, 1
)
3 3 3
Next clear fractions to obtain
𝑤2 = (−2, 1, 1)
Then find
𝑣3 −
<𝑣3|𝑤1> <𝑣3|𝑤2>
‖𝑤 ‖2
1 2
− ‖𝑤 ‖2
= (0, 0, 1) − 1
(1, 1, 1) − 1
(−2, 1,
1)
3 6
= (0, − 1
, 1
)
2 2
Again clear fractions to get,
𝑤3 = (0, −1, 1)
Normally {𝑤1, 𝑤2, 𝑤3} to obtain the following required orthonormal basis of 𝑅3.
{𝑢1 = (1, 1, 1),
1 1
√3
𝑢2 = (−2, 1, 1), 𝑢3 = (0, −1, 1)}
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√6
1
√2