The function f(x, y) and associated matrices e and E

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THE FUNCTION f(x, y) AND ASSOCIATED MATRICES e AND E

The function z = f(x, y) associated with the equation f(x, y) = 0. Associated with the general equation of the second degree

1) ax² + 2hxy + by² + 2gx + 2fy + c = 0

is the function

2) f(x, y) = ax² + 2hxy + by² + 2gx + 2fy + c

to which we now wish to direct our attention.

Let us first make some general observations in connection with the equation

ax² + bx + c = 0

which is the single variable analogue of equation 1) above. Associated with this equation is the function

f(x) = ax² + bx + c

or, equivalently,

y = ax² + bx + c

whose graph is always a parabola which is symmetric about some vertical axis. See Fig. 1. Its trace on the x axis corresponds to the solution set of the equation ax² + b x + c = 0 . A translation of the coordinate system by the distance d in the x direction, so as to position the origin over the parabola axis, will eliminate the term in x giving an equation of the form

y = ax² + q

with the coordinate system then at the position shown in Fig. 2. It can be shown that

The constant term q is equal to the distance of the vertex above origin (the value of y at x = 0). In the figure q is negative.

Now we have a completely analogous situation when we go from an equation in a single variable f(x) = 0 to the equation in two variables

ax² + 2hxy + by² + 2gx + 2fy + c = 0

with which is associated the function

f(x, y) = ax² + 2hxy + by² + 2gx + 2fy + c

or, equivalently,

3) z = ax² + 2hxy + by² + 2gx + 2fy + c .

The function 3) is always a paraboloid (either an elliptic or hyperbolic paraboloid) which is symmetric about some vertical axis in 3-space. See Figures 3 and 4. The equation f(x, y) = 0 corresponds to the trace of z = f(x, y) in the xy-plane. The trace in that plane is one of the 9 conics, a parabola, ellipse, hyperbola, or one of the limiting cases.

A rotation of the coordinate system about the z-axis through the correct angle will always eliminate the xy term in 3) above. What angle? The same angle that eliminates the xy term in the equation f(x, y) = 0. The same rotational transformation that eliminates the xy term in the equation f(x, y) = 0 eliminates the xy term in the parabolic function z = f(x, y).

In the case where f(x, y) = 0 represents a central conic (ellipse, hyperbola) a translation by the proper amounts in the x and y directions will eliminate the terms in x and y of 3) above.

What translation will do this? The same translation that will eliminate the x and y terms in the equation

ax² + 2hxy + by² + 2gx + 2fy + c = 0 .

Intuitively, this amounts to translating the coordinate system in such a way as to position its origin on the axis of the paraboloid. The terms in x and y disappear and our function takes the form

z = ax² + 2hxy + by² + q .

Our coordinate system is then in the position shown in Fig. 5. The constant term q is equal to the distance of the vertex above origin (in the figure q is negative).

In general, when we do the translations and rotations to eliminate the x, y and xy terms in the equation f(x, y) = 0 it can be helpful to think of what is happening in terms of the parabolic function z = f(x, y).

Matrices e and E. The function

f(x, y) = ax² + 2hxy + by² + 2gx + 2fy + c

can also be written in the form

or in the form

4) f(x, y) = (ax + hy + g)x + (hx + by + f)y + (gx + fy + c)

Associated with f(x, y) are two symmetric matrices

which will be seen to come directly from representation 3) above. These two matrices play an central role in the analysis of the general equation of the second degree. We will refer to them later.

Quadratic form. Of importance in connection with the function f(x, y) is the quadratic form

F(x, y) = ax² + 2hxy + by²

which can be written as

or in matrix form as