7Bilinear forms II

IB Linear Algebra

7.2 Hermitian form

The above result was nice for real vector spaces. However, if

φ

is a bilinear form

on a

C

-vector space

V

, then

φ

(

iv, iv

) =

−φ

(

v, v

). So there can be no good

notion of positive definiteness for complex bilinear forms. To make them work

for complex vector spaces, we need to modify the definition slightly to obtain

Hermitian forms.

Definition

(Sesquilinear form)

.

Let

V, W

be complex vector spaces. Then a

sesquilinear form is a function φ : V × W → C such that

(i) φ(λv

1

+ µv

2

, w) =

¯

λφ(v

1

, w) + ¯µφ(v

2

, w).

(ii) φ(v, λw

1

+ µw

2

) = λφ(v, w

1

) + µφ(vw

2

).

for all v, v

1

, v

2

∈ V , w, w

1

, w

2

∈ W and λ, µ ∈ C.

Note that some people have an opposite definition, where we have linearity

in the first argument and conjugate linearity in the second.

These are called sesquilinear since “sesqui” means “one and a half”, and this

is linear in the second argument and “half linear” in the first.

Alternatively, to define a sesquilinear form, we can define a new complex

vector space

¯

V structure on V by taking the same abelian group (i.e. the same

underlying set and addition), but with the scalar multiplication

C ×

¯

V →

¯

V

defined as

(λ, v) 7→

¯

λv.

Then a sesquilinear form on

V × W

is a bilinear form on

¯

V × W

. Alternatively,

this is a linear map W →

¯

V

∗

.

Definition

(Representation of sesquilinear form)

.

Let

V, W

be finite-dimensional

complex vector spaces with basis (

v

1

, ··· , v

n

) and (

w

1

, ··· , w

m

) respectively,

and

φ

:

V × W → C

be a sesquilinear form. Then the matrix representing

φ

with respect to these bases is

A

ij

= φ(v

i

, w

j

).

for 1 ≤ i ≤ n, 1 ≤ j ≤ m.

As usual, this determines the whole sesquilinear form. This follows from

the analogous fact for the bilinear form on

¯

V × W → C

. Let

v

=

P

λ

i

v

i

and

W =

P

µ

j

w

j

. Then we have

φ(v, w) =

X

i,j

λ

i

µ

j

φ(v

i

, w

j

) = λ

†

Aµ.

We now want the right definition of symmetric sesquilinear form. We cannot

just require

φ

(

v, w

) =

φ

(

w, v

), since

φ

is linear in the second variable and

conjugate linear on the first variable. So in particular, if

φ

(

v, w

)

6

= 0, we have

φ(iv, w) 6= φ(v, iw).

Definition

(Hermitian sesquilinear form)

.

A sesquilinear form on

V × V

is

Hermitian if

φ(v, w) = φ(w, v).

Note that if

φ

is Hermitian, then

φ

(

v, v

) =

φ(v, v) ∈ R

for any

v ∈ V

. So

it makes sense to ask if it is positive or negative. Moreover, for any complex

number λ, we have

φ(λv, λv) = |λ|

2

φ(v, v).

So multiplying by a scalar does not change the sign. So it makes sense to talk

about positive (semi-)definite and negative (semi-)definite Hermitian forms.

We will prove results analogous to what we had for real symmetric bilinear

forms.

Lemma.

Let

φ

:

V × V → C

be a sesquilinear form on a finite-dimensional

vector space over

C

, and (

e

1

, ··· , e

n

) a basis for

V

. Then

φ

is Hermitian if and

only if the matrix A representing φ is Hermitian (i.e. A = A

†

).

Proof. If φ is Hermitian, then

A

ij

= φ(e

i

, e

j

) = φ(e

j

, e

i

) = A

†

ij

.

If A is Hermitian, then

φ

X

λ

i

e

i

,

X

µ

j

e

j

= λ

†

Aµ = µ

†

A

†

λ = φ

X

µ

j

e

j

,

X

λ

j

e

j

.

So done.

Proposition

(Change of basis)

.

Let

φ

be a Hermitian form on a finite di-

mensional vector space

V

; (

e

1

, ··· , e

n

) and (

v

1

, ··· , v

n

) are bases for

V

such

that

v

i

=

n

X

k=1

P

ki

e

k

;

and

A, B

represent

φ

with respect to (

e

1

, . . . , e

n

) and (

v

1

, ··· , v

n

) respectively.

Then

B = P

†

AP.

Proof. We have

B

ij

= φ(v

i

, v

j

)

= φ

X

P

ki

e

k

,

X

P

`j

e

`

=

n

X

k,`=1

¯

P

ki

P

`j

A

k`

= (P

†

AP )

ij

.

Lemma

(Polarization identity (again))

.

A Hermitian form

φ

on

V

is determined

by the function ψ : v 7→ φ(v, v).

The proof this time is slightly more involved.

Proof. We have the following:

ψ(x + y) = φ(x, x) + φ(x, y) + φ(y, x) + φ(y , y)

−ψ(x − y) = −φ(x, x) + φ(x, y) + φ(y, x) − φ(y , y)

iψ(x − iy) = iφ(x, x) + φ(x, y) − φ(y, x) + iφ(y , y)

−iψ(x + iy) = −iφ(x, x) + φ(x, y) − φ(y, x) − iφ(y , y)

So

φ(x, y) =

1

4

(ψ(x + y) − ψ(x − y) + iψ(x − iy) − iψ(x + iy)).

Theorem

(Hermitian form of Sylvester’s law of inertia)

.

Let

V

be a finite-

dimensional complex vector space and

φ

a hermitian form on

V

. Then there

exists unique non-negative integers p and q such that φ is represented by

I

p

0 0

0 −I

q

0

0 0 0

with respect to some basis.

Proof. Same as for symmetric forms over R.