Identifying second degree
7.1 The eigenvalue method
In this section we apply eigenvalue methods to determine the geometrical
nature of the second degree equation
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0,
where not all of a, h, b are zero.
Let A =
be the matrix of the quadratic form ax2 +2hxy+ by2.
We saw in section 6.1, equation 6.2 that A has real eigenvalues λ1 and λ2,
a + b −
(a − b)2 + 4h2
, λ2 =
a + b +
(a − b)2 + 4h2
We show that it is always possible to rotate the x, y axes to x1, x2 axes whose
positive directions are determined by eigenvectors X1 and X2 corresponding
to λ1and λ2 in such a way that relative to the x1, y1 axes, equation 7.1 takes
a′x2 + b′y2 + 2g′x + 2f ′y + c = 0.
Then by completing the square and suitably translating the x1, y1 axes,
to new x2, y2 axes, equation 7.2 can be reduced to one of several standard
forms, each of which is easy to sketch. We need some preliminary definitions.
CHAPTER 7. IDENTIFYING SECOND DEGREE EQUATIONS
DEFINITION 7.1.1 (Orthogonal matrix) An n × n real matrix P is
called orthogonal if
P tP = In.
It follows that if P is orthogonal, then detP = ±1. For
det (P tP ) = detP t detP = (detP )2,
so (detP )2 =det In = 1. Hence detP = ±1.
If P is an orthogonal matrix with detP = 1, then P is called a proper
THEOREM 7.1.1 If P is a 2× 2 orthogonal matrix with detP = 1, then
cos θ − sin θ
for some θ.
REMARK 7.1.1 Hence, by the discusssion at the beginning of Chapter
6, if P is a proper orthogonal matrix, the coordinate transformation
represents a rotation of the axes, with new x1 and y1 axes given by the
repective columns of P .
Proof. Suppose that P tP = I2, where ∆ =detP = 1. Let
Then the equation
P t = P−1 =
Hence a = d, b = −c and so
where a2 + c2 = 1. But then the point (a, c) lies on the unit circle, so
a = cos θ and c = sin θ, where θ is uniquely d