hyperbolic pairs and basis

hyperbolic pairs and basis

Definition 1

.

Given a

reflexive non-degenerate sesquilinear

form

b

:

V

×

V

→

k

,
a

hyperbolic pair

is a pair

e

,

f

∈

V

such that

b

⁢

(

e

,

e

)

=

0

=

b

⁢

(

f

,

f

)

⁢

and

⁢

b

⁢

(

e

,

f

)

=

1

.

The span of a hyperbolic pair is a hyperbolic line (recall that a line refers to the

projective dimension

thus we have a 2-dimensional

subspace

but a 1-dimensional

projective space

).

Definition 2

.

A

hyperbolic basis

for a

vector space

V

with respect to a reflexive
non-degenerate sesquilinear form

b

is a basis

{

e

1

,

f

1

,

…

,

e

m

,

f

m

}

where

b

⁢

(

e

i

,

e

j

)

=

0

=

b

⁢

(

f

i

,

f

j

)

⁢

and

⁢

b

⁢

(

e

i

,

f

j

)

=

δ

i

⁢

j

.

Thus a hyperbolic basis is a basis composed of hyperbolic pairs. Furthermore,
if

V

has a hyperbolic basis then setting

H

i

=

⟨

e

i

,

f

i

⟩

shows

V

=

H

1

⟂

H

2

⟂

⋯

⟂

H

m

where

X

⟂

Y

=

X

⊕

Y

with the added condition

b

⁢

(

X

,

Y

)

=

0

.

Hyperbolic bases are the foundation of a “

standard basis

” for a vector spaces

V

equipped with a reflexive non-degenerate sesquilinear form.

1

Symmetric pairs

A

symmetric

hyperbolic pair is a hyperbolic pair

e

,

f

for which

b

restricted to

L

=

⟨

e

,

f

⟩

is a

symmetric bilinear form

. This requires the additional condition that

b

⁢

(

e

,

f

)

=

1

=

b

⁢

(

f

,

e

)

.

This means that the form restricted the

hyperbolic line

L

=

⟨

e

,

f

⟩

can be represented by the matrix

[

0

1

1

0

]

.

When

1

/

2

∈

k

we can consider the associated

quadratic form

q

⁢

(

v

)

=

1

2

⁢

b

⁢

(

v

,

v

)

so if

v

=

x

⁢

e

+

y

⁢

f

we arrive at the

polynomial

q

⁢

(

x

⁢

e

+

y

⁢

f

)

=

1

2

⁢

(

[

x

,

y

]

⁢

[

0

1

1

0

]

⁢

[

x

y

]

)

=

x

⁢

y

.

Suppose the field is

ℝ

. Then we can

associate

a graph to the

equations

c

=

q

⁢

(

x

⁢

e

+

y

⁢

f

)

=

x

⁢

y

for any fixed

c

∈

ℝ

. If

c

=

0

then

x

=

0

or

y

=

0

so the graph is the

x

and

y

-axis – also called the
degenerate hyperbola. If

c

≠

0

then

x

≠

0

and so

y

=

c

x

.
This is the graph of a a hyperbola, hence the title of a hyperbolic pair.

Symmetric

bilinear maps

are often preferred to be presented as

diagonal
matrices

so that they

reflect

the content of Sylvester’s Law of Inertia.
When

1

/

2

∈

k

(

characteristic

of

k

is not 2) we can

diagonalize

any symmetric hyperbolic pair as follows:

[

1

2

1

-

1

2

1

]

⁢

[

0

1

1

0

]

⁢

[

1

2

-

1

2

1

1

]

=

[

1

0

0

-

1

]

.

That is, we can change the basis to

e

↦

u

:=

1

2

⁢

e

+

f

,

f

↦

v

:=

-

1

2

⁢

e

+

f

.

Then

b

⁢

(

u

,

u

)

=

1

,

b

⁢

(

v

,

v

)

=

-

1

, and

b

⁢

(

u

,

v

)

=

b

⁢

(

v

,

u

)

=

0

. Alternatively we find
under this basis we have the quadratic form

q

⁢

(

x

⁢

u

+

y

⁢

v

)

=

x

2

-

y

2

which is also
seen as the standard equation of a hyperbola.

If we think of a quadratic form as generalizing norms – that is length, then
we are observing that on a hyperbolic line length is not

Euclidean

, in fact,
as the usual Euclidean length of

(

x

,

y

)

,

x

2

+

y

2

, gets large, the associated hyperbolic length get small:

x

2

-

y

2

may get small, even 0 or

negative

. Thus the curvature of this space is negative (consider the graph of

z

=

x

2

-

y

2

which is a saddle.)

All symmetric hyperbolic pairs are isometric so decomposing a

bilinear form

into
the

radical

plus hyperbolic pairs plus any left over anisotropic complement produces
a standard basis which allows for easy comparison of one symmetric bilinear form to another.

2

Alternating pairs

An

alternating

hyperbolic pair is a hyperbolic pair

e

,

f

for which

b

restricted
to

L

=

⟨

e

,

f

⟩

is an alternating bilinear form. This requires the
additional condition that

b

⁢

(

e

,

f

)

=

1

=

-

b

⁢

(

f

,

e

)

.

This means that the form restricted the hyperbolic line

L

=

⟨

e

,

f

⟩

can be represented by the matrix

[

0

1

-

1

0

]

.

Although we do not associate a quadratic form with an alternating bilinear
(since

b

⁢

(

v

,

v

)

=

0

for all

v

∈

V

) we can still derive the equations of
a hyperbola. Specifically

b

⁢

(

x

⁢

e

,

y

⁢

f

)

=

x

⁢

y

.

So again setting

c

=

b

⁢

(

x

⁢

e

,

y

⁢

f

)

=

x

⁢

y

we observe the various hyperbola graphs.

Alternating hyperbolic pairs cannot be diagonalized as every

element

v

∈

V

has

b

⁢

(

v

,

v

)

=

0

.

All alternating bilinear forms decompose into hyperbolic lines and the radical and
any two alternating hyperbolic lines are isometric and thus simply indicating the number
of hyperbolic pairs in an alternating bilinear form specifies the form uniquely. If we
further insist the form is

non-degenerate

then the

dimension

of the vector space
specifies the form completely.

3

Hermitian pairs

A

is a hyperbolic pair

e

,

f

for which

b

restricted to

L

=

⟨

e

,

f

⟩

is an

Hermitian

bilinear form. This requires the
additional condition that

b

⁢

(

e

,

f

)

=

1

=

b

⁢

(

f

,

e

)

. The associated matrix does not reveal
much

difference

from the symmetric as we still obtain

[

0

1

1

0

]

.

What is different is how the matrix is used to compute the bilinear products:

b

⁢

(

x

1

⁢

e

+

y

1

⁢

f

,

x

2

⁢

e

+

y

2

⁢

f

)

=

[

x

1

,

y

1

]

⁢

[

0

1

1

0

]

⁢

[

x

2

¯

y

2

¯

]

.

So if we compute

b

⁢

(

v

,

v

)

we find:

b

⁢

(

x

⁢

e

+

y

⁢

f

,

x

⁢

e

+

y

⁢

f

)

=

[

x

,

y

]

⁢

[

0

1

1

0

]

⁢

[

x

¯

y

¯

]

=

x

⁢

y

¯

+

y

⁢

x

¯

=

(

x

⁢

y

¯

)

+

(

x

⁢

y

¯

)

¯

.

We see from this that two hyperbolic pairs of Hermitian type need not be isometric
unless we further consider the

automorphism

of the two forms.

4

Characteristic 2

Hyperbolic pairs over fields of characteristic 2 are a special breed because they
are at the same time symmetric and alternating. That is, the form
is the matrix:

[

0

1

1

0

]

=

[

0

1

-

1

0

]

.

Thus the form cannot be diagonalized as it is alternating. Here it is generally
more useful to use a quadratic form then the bilinear form. Unfortunately
because we cannot recover the quadratic form from the bilinear form
on account that

b

⁢

(

v

,

v

)

=

0

, such a quadratic form must be provided externally
from some other method. Thus it is not always feasible.

Title

hyperbolic pairs and basis

Canonical name

HyperbolicPairsAndBasis

Date of creation

2013-03-22 15:50:59

Last modified on

2013-03-22 15:50:59

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

13

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 15A99

Classification

msc 15A63

Synonym

symplectic pair

Defines

hyperbolic pair

Defines

hyperbolic basis

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