Page 248 -

5.7. BILINEAR AND QUADRATIC FORMS 215

5.7.2-2. Bilinear forms in finite-dimensional spaces.

Any bilinear form B(x, y)onan 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 the vectors x and y in the given basis. The matrix B ≡ [b ij ]
of size n × n is called the matrix of the bilinear form in the given basis e 1 , ... , e n .The
bilinear form can also be represented as

T
T
B(x, y)= X BY , X ≡ (ξ 1 , ... , ξ n ), Y T ≡ (η 1 , ... , η n ).
Remark. Any square matrix B ≡ [b ij] can be regarded as a matrix of some bilinear form in a given basis
e 1, ... , e n. If this matrix is symmetric (skew-symmetric), then the bilinear form is symmetric (skew-symmetric).
The rank of a bilinear form B(x, y)on a finite-dimensional linear space L is defined as
the rank of the matrix B of this form in any basis: rank B(x, y)= rank (B).
A bilinear form on a finite dimensional space V is said to be nondegenerate (degenerate)
if its rank is equal to (is less than) the dimension of the space V, i.e., rank B(x, y)=dim V
(rank B(x, y)<dim V).

5.7.2-3. Transformation of the matrix of a bilinear form in another basis.
Suppose that the transition from a basis e 1 , ... , e n to a basis 2 e 1 , ... , 2 e n is determined by
the matrix U ≡ [u ij ]ofsize n × n,i.e.

n

2 e i = u ij e j (i = 1, 2, ... , n).
j=1

THEOREM. The matrices B and B of a bilinear form B(x, y) in the bases e 1 , ... , e n and
2
2 e 1 , ... , 2 e n , respectively, are related by

T
B = U BU.
2

5.7.2-4. Multilinear forms.
A multilinear form on a linear space V is a scalar function B(x 1 , ... , x p )of p arguments
x 1 , ... , x p V, which is linear in each argument for fixed values of the other arguments.
A multilinear form B(x, y)is said to be symmetric if for any two arguments x l and x l ,
we have
B(x 1 , ... , x k , ... , x l , ... , x p )= B(x 1 , ... , x l , ... , x k , ... , x p ).
A multilinear form B(x, y)issaid tobe skew-symmetric if for any two arguments x l and x l ,
we have
B(x 1 , ... , x k , ... , x l , ... , x p )=–B(x 1 , ... , x l , ... , x k , ... , x p ).