proof of Gram-Schmidt orthogonalization procedure

proof of Gram-Schmidt orthogonalization procedure

Note that, while we state the following as a

theorem

for the sake of logical completeness and to establish notation, our

definition

of

Gram-Schmidt orthogonalization

is wholly

equivalent

to that given in the defining entry.

Theorem

.

(Gram-Schmidt Orthogonalization)

Let

{

u

k

}

k

=

1

n

be a basis for an

inner product space

V

with

inner product

<

,

>

. Define

v

1

=

u

1

∣

∣

u

1

∣

∣

and

{

m

k

}

k

=

2

n

recursively by

𝐦

𝐤

=

𝐮

𝐤

-

<

𝐮

𝐤

,

𝐯

𝟏

>

𝐯

𝟏

-

<

𝐮

𝐤

,

𝐯

𝟐

>

𝐯

𝟐

-

…

-

<

𝐮

𝐤

,

𝐯

𝐤

-

𝟏

>

𝐯

𝐤

-

𝟏

⁢

,

(1)

where

v

k

=

m

k

∣

∣

m

k

∣

∣

for

2

≤

k

≤

n

. Then

{

v

k

}

k

=

1

n

is an

orthonormal basis

for

V

.

Proof.

We proceed by

induction

on

n

. In the case

n

=

1

, we suppose

{

𝐮

𝐤

}

k

=

1

n

=

{

𝐮

𝐤

}

k

=

1

1

=

𝐮

𝟏

is a basis for the inner product space

V

. Letting

𝐯

𝟏

=

𝐮

𝟏

∣

∣

𝐮

𝟏

∣

∣

, it is clear that

𝐯

𝟏

∈

Span

⁡

(

𝐮

𝟏

)

, whence it follows that

Span

⁡

(

𝐯

𝟏

)

=

Span

⁡

(

𝐮

𝟏

)

=

V

. Thus

𝐯

𝟏

is an orthonormal basis for

V

, and the result holds for

n

=

1

. Now let

n

≥

1

∈

ℕ

, and suppose the result holds for arbitrary

n

. Let

{

𝐮

𝐤

}

k

=

1

n

+

1

be a basis for an inner product space

V

. By the

inductive hypothesis

we may use

{

𝐮

𝐤

}

k

=

1

n

to construct an

orthonormal set

of vectors

{

𝐯

𝐤

}

k

=

1

n

such that

Span

⁡

(

{

𝐯

𝐤

}

k

=

1

n

)

=

Span

⁡

(

{

𝐮

𝐤

}

k

=

1

n

)

. In accordance with the procedure outlined in the statement of the theorem, let

𝐦

𝐧

+

𝟏

be defined as

𝐮

𝐧

+

𝟏

-

<

𝐮

𝐧

+

𝟏

,

𝐯

𝟏

>

𝐯

𝟏

-

<

𝐮

𝐧

+

𝟏

,

𝐯

𝟐

>

𝐯

𝟐

-

…

-

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐧

>

𝐯

𝐧

=

𝐮

𝐧

+

𝟏

-

∑

i

=

1

n

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐢

>

𝐯

𝐢

⁢

.

First we show that the vectors

𝐯

𝟏

,

𝐯

𝟐

,

…

,

𝐯

𝐧

,

𝐦

𝐧

+

𝟏

are mutually

orthogonal

.
Consider the inner product of

𝐦

𝐧

+

𝟏

with

𝐯

𝐣

for

1

≤

j

≤

n

. By construction, we have

<

𝐦

𝐧

+

𝟏

,

𝐯

𝐣

≥

<

𝐮

𝐧

+

𝟏

-

∑

i

=

1

n

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐢

>

𝐯

𝐢

,

𝐯

𝐣

≥

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐣

>

-

<

∑

i

=

1

n

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐢

>

𝐯

𝐢

,

𝐯

𝐣

>

.

Now since

{

𝐯

𝐤

}

k

=

1

n

is an orthonormal set of vectors, whence

<

𝐯

𝐢

,

𝐯

𝐣

≥

δ

i

⁢

j

, each term in the

summation

on the right-hand side of the preceding equation will vanish except for the term where

i

=

j

. Thus by this and the preceding equation, we have

<

𝐦

𝐧

+

𝟏

,

𝐯

𝐣

≥

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐣

>

-

<

∑

i

=

1

n

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐢

>

𝐯

𝐢

,

𝐯

𝐣

≥

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐣

>

-

≪

𝐮

𝐧

+

𝟏

,

𝐯

𝐣

>

𝐯

𝐣

,

𝐯

𝐣

>

=

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐣

>

-

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐣

>

<

𝐯

𝐣

,

𝐯

𝐣

≥

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐣

>

-

<

𝐮

𝐧

+

𝟏

,

𝐯

𝐣

≥

0

.

Thus

𝐦

𝐧

+

𝟏

is orthogonal to

𝐯

𝐣

for

1

≤

j

≤

n

, so we may take

𝐯

𝐧

+

𝟏

=

𝐦

𝐧

+

𝟏

∣

∣

𝐦

𝐧

+

𝟏

∣

∣

to have

{

𝐯

𝐤

}

k

=

1

n

+

1

an orthonormal set of vectors. Finally we show that

{

𝐯

𝐤

}

k

=

1

n

+

1

is a basis for

V

.
By construction, each

𝐯

𝐤

is a

linear combination

of the vectors

{

𝐮

𝐤

}

k

=

1

n

+

1

, so we have

n

+

1

orthogonal, hence

linearly independent

vectors in the

n

+

1

dimensional space

V

, from which it follows that

{

𝐯

𝐤

}

k

=

1

n

+

1

is a basis for

V

. Thus the result holds for

n

+

1

, and by the principle of induction, for all

n

.
∎

Title

proof of Gram-Schmidt orthogonalization procedure

Canonical

name

ProofOfGramSchmidtOrthogonalizationProcedure

Date of creation

2013-03-22 16:27:17

Last modified on

2013-03-22 16:27:17

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

11

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 65F25

Synonym

Gram-Schmidt

Synonym

orthogonalization

Related topic

GramSchmidtOrthogonalization

Related topic

ExampleOfGramSchmidtOrthogonalization

Related topic

InnerProductSpace

Related topic

InnerProduct

Related topic

QRDecomposition

Related topic

NormedVectorSpace

Related topic

Orthogonal

Related topic

OrthogonalVectors

Related topic

Basis

Related topic

Span

Related topic

LinearIndependence

Related topic

Orthonormal

Related topic

OrthonormalBasis

Defines

Gram-Schmidt orthogonalization

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