Page 276 - Advanced Linear Algebra

260 Advanced Linear Algebra

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3 alternate ( or alternating ) if
º%Á %» ~

for all %= .
A bilinear form that is either symmetric, skew-symmetric, or alternate is
referred to as an inner product and a pair ²= ÁºÁ»³ , where is a vector space
=
=
and ºÁ » is an inner product on , is called a metric vector space or inner
product space. As usual, we will refer to as a metric vector space when the
=
form is understood.
4 A metric vector space with a symmetric form is called an orthogonal
)
=
geometry over .
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5 A metric vector space = with an alternate form is called a symplectic
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geometry over .
The term symplectic , from the Greek for “intertwined,” was introduced in 1939
by the famous mathematician Hermann Weyl in his book The Classical Groups ,
as a substitute for the term complex . According to the dictionary, symplectic
means “relating to or being an intergrowth of two different minerals.” An
example is ophicalcite , which is marble spotted with green serpentine.

Example 11.1 Minkowski space 4 4 is the four-dimensional real orthogonal
geometry s with inner product defined by
º Á » ~ º Á » ~ º Á » ~
3
3

º Á » ~ c
4
4
º Á » ~ for £

where Á Ã Á 4 is the standard basis for s .

As is traditional, when the inner product is understood, we will use the phrase
“let be a metric vector space.”
=
The real inner products discussed in Chapter 9 are inner products in the present
sense and have the additional property of being positive definite —a notion that
does not even make sense if the base field is not ordered. Thus, a real inner
product space is an orthogonal geometry. On the other hand, the complex inner
products of Chapter 9, being sesquilinear, are not inner products in the present
sense. For this reason, we use the term metric vector space in this chapter, rather
than inner product space .
If is a vector subspace of a metric vector space , then inherits the metric
:
=
:
structure from . With this structure, we refer to as a subspace of .
:
=
=
The concepts of being symmetric, skew-symmetric and alternate are not
independent. However, their relationship depends on the characteristic of the
base field , as do many other properties of metric vector spaces. In fact, the
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