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218 ALGEBRA

The basis e 1 , ... , e n is transformed to the canonical basis g 1 , ... , g n by the formulas
n

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

α ij =(–1) i+j Δ i–1,j ,
Δ i–1
where Δ i–1,j is the minor of the submatrix of B ≡ [b ij ] formed by the elements on the
intersection of its rows with indices 1, 2, ... , i – 1 and columns with indices 1, 2, ... , j – 1,
j + 1, i.

5.7.3-5. Normal representation of a real quadratic form.

Let g 1 , ... , g n be a basis of a linear space V in which the quadratic form B(x, x) is written as
n
2
B(x, x)= ε i η , (5.7.3.3)
i
i=1
where η 1 , ... , η n are the coordinates of x in that basis, and ε 1 , ... , ε n are coefficients
taking the values –1, 0,or 1. Such a representation of a quadratic form is called its normal
representation.
Any real quadratic form B(x, x)in an n-dimensional real linear space V admits a
normal representation (5.7.3.3). Such a representation can be obtained by the following
transformations:
1. One obtains its canonical representation (see Paragraph 5.7.3-4):

n
2
B(x, x)= λ i μ .
i
i=1
2. By the nondegenerate coordinate transformation
1
⎧
⎪ √ μ i for λ i > 0,
⎨
λ i
η i = √ 1 μ i for λ i < 0,
–λ i
⎪
⎩
μ i for λ i = 0,
the canonical representation turns into a normal representation.
LAW OF INERTIA OF QUADRATIC FORMS. The number of terms with positive coefficients
and the number of terms with negative coefficients in any normal representation of a real
quadratic form does not depend on the method used to obtain such a representation.
The index of inertia of a real quadratic form is the integer k equal to the number of
nonzero coefficients in its canonical representation (this number coincides with the rank
of the quadratic form). Its positive index of inertia is the integer p equal to the number
of positive coefficients in the canonical representation of the form, and its negative index
of inertia is the integer q equal to the number of its negative canonical coefficients. The
integer s = p – q is called the signature of the quadratic form.
A real quadratic form B(x, x)on an n-dimensional real linear space V is
a) positive definite (resp., negative definite) if p = n (resp., q = n);
b) alternating if p ≠ 0, q ≠ 0;
c) nonnegative (resp., nonpositive) if q = 0, p < n (resp., p = 0, q < n).