The inner product is an example of a bilinear function that likely is familiar to the reader. In this book, we are more interested in a less restrictive type of bilinear function called a scalar product, but we will not ignore the inner product altogether. In order to decide whether a bilinear function is a scalar product, we need to determine whether it is symmetric and nondegenerate. These properties of bilinear functions will be the focus of the present chapter.
3.1 Bilinear Functions
Let V be a vector space. A function
is said to be bilinear (on V) if it is linear in both arguments; that is,
and
for all vectors u, v, w in V and all real numbers c. In the literature, a bilinear function is sometimes called a quadratic form. We often denote
writing 〈v, w〉 in place of b(v, w).
We say that b is:
symmetric
if 〈v, w〉 = 〈w, v〉 for all v, w in V.
alternating
if 〈v, w〉 = − 〈w, v〉 for all v, w in V.
nondegenerate
if for all v in V, 〈v, w〉 = 0 for all w in V implies v = 0.
degenerate
if b is not nondegenerate.
positive definite
if 〈v, v〉 > 0 for all nonzero v in V.
negative definite
if 〈v, v〉 < 0 for all nonzero v in V.
definite
if b is either positive definite or negative definite.
indefinite
if b is not definite.
positive semidefinite
if 〈v, v〉 ≥ 0 for all v in V.
negative semidefinite
if 〈v, v〉 ≤ 0 for all v in V.
semidefinite
if b is either positive semidefinite or negative semidefinite.
Let U be a subspace of V. For brevity, we denote the restriction
In fact, we often drop “|U” from the notation altogether and say, for example, that b is symmetric on U if b|U is symmetric. Evidently, b is bilinear on U, and if b is symmetric (resp., alternating, definite, semidefinite) on V, then b is symmetric (resp., alternating, definite, semidefinite) on U. The situation is more complicated with nondegeneracy. If b is nondegenerate on V, it does not necessarily follow that b is nondegenerate on U. We will explore this phenomenon later on.
We close this section with a few definitions that will be useful later on. Let V be a vector space, and let b be a bilinear function on V. The quadratic function corresponding to b is the function
Several of the earlier definitions can be expressed in terms of q. Specifically, b is:
positive definite
if q(v) > 0 for all nonzero v in V.
negative definite
if q(v) < 0 for all nonzero v in V.
positive semidefinite
if q(v) ≥ 0 for all v in V.
negative semidefinite
if q(v) ≤ 0 for all v in V.
spacelike
if v = 0 or q(v) > 0.
timelike
if q(v) < 0.
lightlike
if v ≠ 0 and q(v) = 0.
We say that v is a unit vector if 〈v, v〉 = ± 1, and that v is orthogonal to w if 〈v, w〉 = 0. In particular, the zero vector is orthogonal to every vector in V, and a lightlike vector is orthogonal to itself. Unless b is symmetric, 〈w, v〉 may not equal 〈v, w〉. Thus, even when v is orthogonal to w, it does not necessarily follow that w is orthogonal to v.
Let U be a subspace of V. The perp of U (in V with respect to b) is denoted by U⊥ and defined to be the set of vectors in V that are orthogonal (perpendicular) to every vector in U:
It is easily shown that U⊥ is a subspace of V. Let us denote
The light set of V is denoted by Λ and defined to be the set of lightlike vectors in V:
3.2 Symmetric Bilinear Functions
Let V be a vector space, and let b be a bilinear function on V. Recall from Section 3.1 that b is said to be symmetric on V if 〈v, w〉 = 〈w, v〉 for all vectors v, w in V.
Theorem 3.2.2 shows that provided b is symmetric, b and q determine each other completely.
We are especially interested in vector spaces on which there is a nonzero symmetric bilinear function. The reason is that such vector spaces have a unit vector, as we now show.
The light set of a vector space is a rather mysterious entity. The next result provides a characterization when the bilinear function satisfies certain properties.
The next result is just one of the many versions of the Cauchy–Schwarz inequality that arise in several areas of mathematics. We will see another, likely more familiar, variation in Section 4.6.
3.3 Flat Maps and Sharp Maps
Readers acquainted with tensor analysis, especially in the context of theoretical physics, may have some familiarity with computations involving “lowering” and “raising” indices. This section and Section 4.5 develop these ideas using flat maps and sharp maps.
Let V be a vector space, and let b be a bilinear function on V. The flat map corresponding to b is the linear map
defined by
for all vectors v, w in V. We usually denote
so that
(3.3.1)
We see from (3.3.2)–(3.3.4) why taking the flat of v to obtain vF is classically referred to as lowering an index by b.
Let V be a vector space, and let b be a bilinear function on V. Recall from Section 3.1 that b is said to be nondegenerate on V if for all vectors v in V, 〈v, w〉 = 0 for all vectors w in V implies v = 0.
Let V be a vector space, and let b be a nondegenerate bilinear function on V. By Theorem 3.3.3, F is a linear isomorphism. Its inverse, also a linear isomorphism according to Theorem 1.1.9, is denoted by
and called the sharp map corresponding to b. Thus,
(3.3.5)
We usually denote
for all covectors η in V*. In much of the mathematical literature, F and S are represented by the flat and sharp music symbols, b and #. For our purposes the present notation is more convenient.
Let us observe that with the identification V** = V, the pullback by F : V → V* and the pullback by S : V* → V are expressed as
