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5.7. BILINEAR AND QUADRATIC FORMS 217

5.7.3-4. Canonical representation of a real quadratic form.

Let g 1 , ... , g n be a basis in which the real quadratic form B(x, x) in a linear space V admits
the representation
n
2
B(x, x)= λ i η , (5.7.3.2)
i
i=1
where η 1 , ... , η n are the coordinates of x in that basis. This representation is called a
canonical representation of the quadratic form, the real coefficients λ 1 , ... , λ n are called
the canonical coefficients, and the basis g 1 , ... , g n is called the canonical basis.
The number of nonzero canonical coefficients is equal to the rank of the quadratic form.
THEOREM. Any real quadratic form on an n-dimensional real linear space V admits a
canonical representation (5.7.3.2).

1 . Lagrange method. The basic idea of the method consists of consecutive transformations
◦
of the quadratic form: on every step, one should single out the perfect square of some linear
form.
Consider a quadratic form
n

B(x, x)= b ij ξ i ξ j .
i,j=1

Case 1. Suppose that for some m (1 ≤ m ≤ n), we have b mm ≠ 0. Then, letting
1 n 2

B(x, x)= b mk ξ k + B 2 (x, x),
b mm
k=1
one can easily verify that the quadratic form B 2 (x, x) does not contain the variable ξ m .
This method of separating a perfect square in a quadratic form can always be applied if the
matrix [b ij ](i, j = 1, 2, ... , n) contains nonzero diagonal elements.
Case 2. Suppose that b mm = 0, b ss = 0,but b ms ≠ 0. In this case, the quadratic form
can be represented as

1 n 2 1 n 2

B(x, x)= (b mk + b sk )ξ k – (b mk – b sk )ξ k + B 2 (x, x),
2b sm 2b sm
k=1 k=1
where B 2 (x, x) does not contain the variables ξ m , ξ s , and the linear forms in square brackets
are linearly independent (and therefore can be taken as new independent variables or
coordinates).
By consecutive combination of the above two procedures, the quadratic form B(x, x) can
always be represented in terms of squared linear forms; these forms are linearly independent,
since each contains a variable which is absent in the other linear forms.
2 . Jacobi method. Suppose that
◦

b 11 b 12
Δ 1 ≡ b 11 ≠ 0, ≠ 0, ... , Δ n ≡ det B ≠ 0,
Δ 2 ≡
b 21 b 22
where B ≡ [b ij ] is the matrix of the quadratic form B(x, x)in some basis e 1 , ... , e n .One
can obtain a canonical representation of this form using the formulas
Δ i
λ 1 = Δ 1 , λ i = (i = 2, 3, ... , n).
Δ i–1