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216 ALGEBRA

5.7.3. Quadratic Forms
5.7.3-1. Definition of a quadratic form.
A quadratic form on a real linear space is a scalar function B(x, x) obtained from a bilinear
form B(x, y)for x = y.
Any symmetric bilinear form B(x, y)is polar with respect to the quadratic form B(x, x).
These forms are related by
1
B(x, y)= [B(x + y, x + y)– B(x, x)– B(y, y)].
2

5.7.3-2. Quadratic forms in a finite-dimensional linear space.

Any quadratic form B(x, x)inan n-dimensional linear space with a given basis e 1 , ... , e n
can be uniquely represented in the form
n

B(x, x)= b ij ξ i ξ j , (5.7.3.1)
i,j=1
where ξ i are the coordinates of the vector x in the given basis, and B ≡ [b ij ] is a symmetric
matrix of size n × n, called the matrix of the bilinear form B(x, x) in the given basis. This
quadratic form can also be represented as
T
T
B(x, x)= X BX, X ≡ (ξ 1 , ... , ξ n ).
Remark. Any quadratic form can be represented in the form (5.7.3.1) with infinitely many matrices B
T
such that B(x, x)= X BX. In what follows, we consider only one of such matrices, namely, the symmetric
matrix. A quadratic form is real-valued if its symmetric matrix is real.
A real-valued quadratic form B(x, x) is said to be:
a) positive definite (negative definite)if B(x, x)> 0 (B(x, x)< 0)for any x ≠ 0;
b) alternating if there exist x and y such that B(x, x)> 0 and B(y, y)< 0;
c) nonnegative (nonpositive)if B(x, x) ≥ 0 (B(x, x) ≤ 0)for all x.
If B(x, y) is a polar bilinear form with respect to some positive definite quadratic form
B(x, x), then B(x, y) satisfies all axioms of the scalar product in a Euclidean space.
Remark. The axioms of the scalar product can be regarded as the conditions that determine a bilinear
form that is polar to some positive definite quadratic form.
The rank of a quadratic form on a finite-dimensional linear space V is, by definition,
the rank of the matrix of that form in any basis of V,rank B(x, x)=rank (B).
A quadratic form on a finite-dimensional linear space V is said to be nondegenerate
(degenerate) if its rank is equal to (is less than) the dimension of V, i.e., rank B(x, x)= dim V
(rank B(x, x)<dim V).

5.7.3-3. Transformation of a bilinear form in another basis.

Suppose that the transition from the basis e 1 , ... , e n to the basis 2 e 1 , ... , 2 e n is given by the
matrix U ≡ [u ij ]of size n × n,i.e.
n

2 e i = u ij e j (i = 1, 2, ... , n).
j=1
Then the matrices B and B of the quadratic form B(x, x) in the bases e 1 , ... , e n and
2
2 e 1 , ... , 2 e n , respectively, are related by
T
B = U BU.
2