Properties of Vector Space Inner Products: Addition, Subtraction, and Scaling

Vector spaces equipped with an inner product form a fundamental structure in linear algebra. The inner product satisfies several important properties such as bilinearity, symmetry, and positive definiteness. In this article, we will analyze whether the sum, difference, or positive multiple of two inner products maintains these properties and is itself an inner product.

Bilinear Nature of Inner Products

To understand the properties of inner products, let's first recall that an inner product on a vector space ( V ) over a field ( F ) must satisfy the following properties:

Bilinearity

Bilinearity over the first argument: ( langle u v, w rangle langle u, w rangle langle v, w rangle )

Bilinearity over the second argument: ( langle u, v w rangle langle u, v rangle langle u, w rangle )

Homogeneity: If ( c in F ), then ( langle cu, v rangle c langle u, v rangle )

Symmetry (or Conjugate Symmetry)

In the real field, symmetry: ( langle u, v rangle langle v, u rangle )

In the complex field, conjugate symmetry: ( langle u, v rangle overline{langle v, u rangle} )

Positive Definiteness

( langle v, v rangle geq 0 ) for all ( v in V )

( langle v, v rangle 0 ) if and only if ( v 0 )

Sum of Two Inner Products

Let ( langle cdot, cdot rangle_1 ) and ( langle cdot, cdot rangle_2 ) be two inner products on ( V ). We define a new inner product ( langle cdot, cdot rangle_s ) as follows:

[ langle u, v rangle_s langle u, v rangle_1 langle u, v rangle_2 ]

Let's check if ( langle cdot, cdot rangle_s ) satisfies the properties of an inner product:

Bilinearity

( langle u v, w rangle_s langle u v, w rangle_1 langle u v, w rangle_2 langle u, w rangle_1 langle v, w rangle_1 langle u, w rangle_2 langle v, w rangle_2 langle u, w rangle_s langle v, w rangle_s )

( langle u, v w rangle_s langle u, v w rangle_1 langle u, v w rangle_2 langle u, v rangle_1 langle u, w rangle_1 langle u, v rangle_2 langle u, w rangle_2 langle u, v rangle_s langle u, w rangle_s )

( langle cu, v rangle_s c langle u, v rangle_1 c langle u, v rangle_2 c (langle u, v rangle_1 langle u, v rangle_2) c langle u, v rangle_s )

Symmetry

( langle u, v rangle_s langle u, v rangle_1 langle u, v rangle_2 overline{langle v, u rangle_1} overline{langle v, u rangle_2} overline{langle v, u rangle_s} )

Positive Definiteness

( langle v, v rangle_s langle v, v rangle_1 langle v, v rangle_2 geq 0 )

For ( langle v, v rangle_s 0 ), both ( langle v, v rangle_1 0 ) and ( langle v, v rangle_2 0 ). Since each inner product is positive definite, ( v 0 ).

Hence, the sum of two inner products is indeed an inner product.

Difference of Two Inner Products

Let's consider the difference:

[ langle u, v rangle_d langle u, v rangle_1 - langle u, v rangle_2 ]

Checking the properties:

Bilinearity

( langle u v, w rangle_d langle u v, w rangle_1 - langle u v, w rangle_2 langle u, w rangle_1 langle v, w rangle_1 - langle u, w rangle_2 - langle v, w rangle_2 langle u, w rangle_d langle v, w rangle_d )

( langle u, v w rangle_d langle u, v w rangle_1 - langle u, v w rangle_2 langle u, v rangle_1 langle u, w rangle_1 - langle u, v rangle_2 - langle u, w rangle_2 langle u, v rangle_d langle u, w rangle_d )

( langle cu, v rangle_d c langle u, v rangle_1 - c langle u, v rangle_2 c (langle u, v rangle_1 - langle u, v rangle_2) c langle u, v rangle_d )

Symmetry

( langle u, v rangle_d langle u, v rangle_1 - langle u, v rangle_2 overline{langle v, u rangle_1} - overline{langle v, u rangle_2} - overline{- langle u, v rangle_1} overline{langle v, u rangle_2} - overline{langle u, v rangle_2} overline{langle v, u rangle_1} overline{langle v, u rangle_d} )

Positive Definiteness

( langle v, v rangle_d langle v, v rangle_1 - langle v, v rangle_2 )

The value can be zero even for non-zero ( v ) if ( langle v, v rangle_1 langle v, v rangle_2 ), violating positive definiteness. For example, if ( langle v, v rangle_1 langle v, v rangle_2 ) for some non-zero ( v ), then ( langle v, v rangle_d 0 ).

Hence, the difference of two inner products is not necessarily an inner product.

Positive Multiple of an Inner Product

Let ( c in F ) be a positive scalar. We define:

[ langle u, v rangle_p c langle u, v rangle ]

Let's check if ( langle cdot, cdot rangle_p ) satisfies the properties of an inner product:

Bilinearity

( langle u v, w rangle_p c langle u v, w rangle c (langle u, w rangle langle v, w rangle) c langle u, w rangle c langle v, w rangle langle u, w rangle_p langle v, w rangle_p )

( langle u, v w rangle_p c langle u, v w rangle c (langle u, v rangle langle u, w rangle) c langle u, v rangle c langle u, w rangle langle u, v rangle_p langle u, w rangle_p )

( langle cu, v rangle_p c langle cu, v rangle c (c langle u, v rangle) c langle u, v rangle_p )

Symmetry

( langle u, v rangle_p c langle u, v rangle c overline{langle v, u rangle} overline{c langle v, u rangle} overline{langle v, u rangle_p} )

Positive Definiteness

( langle v, v rangle_p c langle v, v rangle )

Since ( c > 0 ) and ( langle v, v rangle geq 0 ), ( langle v, v rangle_p geq 0 ). Moreover, if ( langle v, v rangle_p 0 ), then ( c langle v, v rangle 0 ). Because ( c eq 0 ), ( langle v, v rangle 0 ), which implies ( v 0 ).

Hence, a positive multiple of an inner product is indeed an inner product.

Summary

The sum of two inner products is an inner product.

The difference of two inner products is not necessarily an inner product.

A positive multiple of an inner product is an inner product.

Understanding these properties and operations on inner products is crucial for various applications in mathematics and physics, particularly in Hilbert spaces and functional analysis.