QUADRATIC FORM IN TWO VARIABLES

Quadratic form in two variables. The function

1)        F(x, y) = ax² + 2bxy + cy²

is a quadratic form in two variables. The coefficients a, b, c may be any constants.

Associated with the quadratic form F(x, y) = ax² + 2bxy + cy² is matrix e

Discriminant of a quadratic form. The discriminant of the quadratic form F(x, y) = ax² + 2bxy + cy² is the determinant of matrix e

Family of curves associated with a quadratic form. Associated with a quadratic form F(x, y) = ax² + 2bxy + cy² is a family of curves

3)        F(x, y) = k

where k is a parameter. These curves will be either ellipses, hyperbolas, or degenerate forms thereof, centered at the origin. If Δ > 0, the curves are ellipses. If Δ < 0, the curves are hyperbolas except in the special case where k = 0, when the locus is the pair of lines which are the asymptotes of all the hyperbolas in the family. If Δ = 0, the family is a an assemblage of parallel straight lines.

Invariance of the discriminant Δ upon a rotation of axes. Let a quadratic form F(x, y) = ax² + 2bxy + cy² be subjected to the change of variables

            x = x' cos θ - y' sin θ

            y = x' sin θ + y' cos θ ,

a transformation that corresponds to a rotation of the coordinate system about the origin by an angle θ, to give

4)        G(x', y') = a'x' ² + 2b'x'y' + c'y^{' 2} . 

Under such a transformation the discriminant Δ is invariant. In other words, under a rotation transformation, the discriminant of G(x', y') is the same as that of F(x, y) [ i.e. a'c' - b^{' 2} = ac - b² ] .

Theorem 1. Any quadratic form can be transformed by a suitably chosen rotation of the coordinate system into a form that contains no term in xy. The rotation angle that will eliminate the xy term is given by

Theorem 2. If a quadratic form F(x, y) = ax² + 2bxy + cy² is subjected to the change of variables

            x = x' cos θ - y' sin θ

            y = x' sin θ + y' cos θ ,

where θ corresponds to that rotation of the coordinate system which eliminates the xy term, thus transforming F(x, y) into

            G(x', y') = a'x' ² + c'y^{' 2} ,

then the coefficients a' and c' correspond to the eigenvalues of matrix e. That is, they correspond to the roots of the characteristic equation of matrix e, the roots of equation

[in expanded form the characteristic equation is λ² - (a + c)λ + ac - b² = 0 ].

Proof. Consider the following quadratic form (which we have pulled from our magic hat)

7)        P(x, y) = F(x, y) - λ(x² + y²)

                        = ax² + 2bxy + cy² - λ(x² + y²)

                        = (a - λ)x² + 2bxy + (c - λ)y² .

We note two things about this function:

1. If P(x, y) is subjected to a transformation consisting of a rotation of the coordinate system about its origin by an angle θ, then substitution, expansion, etc. will show that it will transform into Q(x', y') where

8)        Q(x', y') = G(x', y') - λ(x' ² + y' ²) = (a' - λ)x' ² + 2b'x'y' + (c' - λ)y^{' 2}

[ G(x', y') is the G(x', y') of 4) above].

2. That angle of rotation θ that will eliminate the xy term in P(x, y) is given by

which is the same angle that will eliminate the xy term in F(x, y).

Let us now rotate the coordinate system by an angle θ chosen so as to eliminate the xy term. P(x, y) will then transform into

9)        Q(x', y') = (a' - λ)x' ² + (c' - λ)y^{' 2} .

Due to the invariance of the discriminant Δ, the discriminant of P(x, y) is equal to the discriminant of Q(x', y') i.e.

From inspection of 10) we can see that λ = a' and λ = c' are the roots of the equation

End of proof.

We thus see that the characteristic equation of matrix e

is obtained by setting equal to zero the discriminant of the form

            F(x, y) - λ(x² + y²) = (a - λ)x² + 2bxy + (c - λ)y² .

Reference

  Taylor. Advanced Calculus. p. 205, 206