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214 ALGEBRA

Remark. Thus, B(x, y) is a scalar function that is linear with respect to its first argument and antilinear
with respect to its second argument. For a real space V, sesquilinear forms turn into bilinear forms (see
Paragraph 5.7.2).
THEOREM. Let B(x, y) be a sesquilinear form in a unitary space V. Then there is a
unique linear operator A in L(V, V) such that
B(x, y)= x ⋅ (Ay).
COROLLARY. If B(x, y) is a sesquilinear form in a unitary space V , then there is a unique
linear operator A in L(V, V) such that
B(x, y)=(Ax) ⋅ y.

5.7.1-3. Matrix of a sesquilinear form.
Any sesquilinear form B(x, y)on an n-dimensional linear space with a given basis e 1 , ... ,
e n can be uniquely represented as
n

B(x, y)= b ij ξ i ¯η j , b ij = B(e i , e j ),
i,j=1

and ξ i , η j are the coordinates of x and y in the given basis. The matrix B ≡ [b ij ]ofsize
n × n is called the matrix of the sesquilinear form B(x, y) in the given basis e 1 , ... , e n .
This sesquilinear form can also be represented as
T
T
B(x, y)= X BY , X ≡ (ξ 1 , ... , ξ n ), Y T ≡ (¯η 1 , ... , ¯η n ).
5.7.2. Bilinear Forms
5.7.2-1. Definition of a bilinear form.

A bilinear form on a real linear space V is a real-valued function B(x, y)oftwo arguments
x L, y V satisfying the following conditions for any vectors x, y,and z in V and any
real λ:
1. B(x + y, z)= B(x, z)+ B(y, z).
2. B(x, y + z)= B(x, y)+ B(x, z).
3. B(λx, y)= B(x, λy)= λB(x, y).
THEOREM. Let B(x, y) be a bilinear form in a Euclidean space V. Then there is a unique
linear operator A in L(V, V) such that
B(x, y)=(Ax) ⋅ y.

A bilinear form B(x, y)is saidtobe symmetric if for any x and y,we have

B(x, y)= B(y, x).
A bilinear form B(x, y)is saidtobe skew-symmetric if for any x and y,we have
B(x, y)=–B(y, x).
Any bilinear form can be represented as the sum of symmetric and skew-symmetric bilinear
forms.
THEOREM. A bilinear form B(x, y) on a Euclidean space V is symmetric if and only if
the linear operator A in the representation (5.6.6.1) is Hermitian (A = A ).
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