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5.7. BILINEAR AND QUADRATIC FORMS 219

5.7.3-6. Criteria of positive and negative definiteness of a quadratic form.

1 . A real quadratic form B(x, x) is positive definite, negative definite, alternating, non-
◦
negative, nonpositive if the eigenvalues λ i of its matrix B ≡ [b ij ] are all positive, are all
negative, some are positive and some negative, are all nonnegative, are all nonpositive,
respectively.

◦
2 . Sylvester criterion. A real quadratic form B(x, x) is positive definite if and only if the
matrix of B(x, x)in somebasis e 1 , ... , e n satisfies the conditions

b 11 b 12

Δ 1 ≡ b 11 > 0, > 0, ... , Δ n ≡ det B > 0.
b 21 b 22
Δ 2 ≡
If the signs of the minor determinants alternate,
Δ 1 < 0, Δ 2 > 0, Δ 3 < 0, ... ,

then the quadratic form is negative definite.

◦
3 . A real matrix B is nonnegative and symmetric if and only if there is a real matrix C
T
such that B = C C.
5.7.4. Bilinear and Quadratic Forms in Euclidean Space

5.7.4-1. Reduction of a quadratic form to a sum of squares.

THEOREM 1. Let B(x, y) be a symmetric bilinear form on a n-dimensional Euclidean
space V. Then there is an orthonormal basis i 1 , ... , i n in V and there are real numbers λ k
such that for any x V the real quadratic form B(x, x) can be represented as the sum of
squares of the coordinates ξ k of x in the basis i 1 , ... , i n :

n
2
B(x, x)= λ k ξ .
k
k=1
THEOREM 2. Let A(x, y) and B(x, y) be symmetric bilinear forms in a n-dimensional
real linear space V, and suppose that the quadratic form A(x, x) is positive definite. Then
there is a basis i 1 , ... , i n of V such that the quadratic forms A(x, x) and B(x, x) can be
represented in the form

n n
2 2
A(x, x)= λ k ξ , B(x, x)= ξ ,
k
k
k=1 k=1
where ξ k are the coordinates of x in the basis i 1 , ... , i n . The set of real λ 1 , ... , λ n coincides
–1
with the spectrum of eigenvalues of the matrix B A (the matrices A and B can be taken
in any basis), and this set consists of the roots of the algebraic equation

det(A – λB)= 0.