**Quadratics in the plane**

** **

__ __

__ __

__ __

This application uses the notions of eigenvalues, eigenvectors, and the process of matrix diagonalization. We refer the reader to the application to Fibonacci numbers where these notions were reviewed.

__ __

__ __

** Definition** In the
plane (

_{}

where *A, B* and *C *are
real numbers. The above form can also be written using matrix multiplication:

_{}

or

_{}

where

_{}.

Note that *A* is a symmetric matrix.

** Example** Consider the quadratic form

_{}

which can be written as

_{}

We would like to give a geometric description of the plane curve given by the above equation. To this end, we “diagonalize” the matrix

_{}

The characteristic polynomial of *A* is

_{}

The eigenvalues of A are then 1/2 and 3/2, and one can show that the vector

_{}

is a basis for the eigenspace of *A *corresponding to
the eigenvalue 3/2 and that

_{}

is a basis for the eigenspace of *A *corresponding to
the eigenvalue 1/2. Let

_{}

(the constant in front of the matrix is there to make its
columns of length 1) then _{} where

_{}

is the “diagonal form” of *A*. Note that _{} so equation (*) above
becomes

_{}

But

_{}

and

_{}

So,

_{}.

Now, let us consider the following change of variables:

_{}

so the above equation can be written as:

_{}

In the new system _{}, the equation of the curve is:

_{}

This is clearly the equation of an ellipse with axes √(2/3) and √2.

We conclude that

_{}

is the equation of a rotated ellipse with foci (1, 1), (-1, 1) and axes √(2/3), √2.

Graphically, the following diagram represents the curve:

Another question that is important to answer in this example
is what is the rotation angle that will get rid of the product *xy* in the
curve equation

_{}?

To answer this question, we look again at the “orthogonal” matrix

_{}

(orthogonal in the sense that *P ^{-1 }=P^{T}*).
Note that

_{},

so *P *can be
written as

_{}

with θ=π/4. Such a matrix is called a **rotation
matrix** and is always orthogonal. So, if one makes a rotation of 45^{ο
}of the original axes, one obtains an ellipse in “standard” form in the new system.

Let us look at another example.

** Example **Consider The quadratic form

_{}

then its corresponding matrix is

_{}

The eigenvalues of A are

_{}.

One can easily verify that the vector

_{}

is a basis for the eigenspace of *A *corresponding to
the eigenvalue √5/2 and that

_{}

is a basis for the eigenspace of *A *corresponding to
the eigenvalue -√5/2. Let

_{}

be the rotation matrix in this case, then with respect to a
new system *(x _{1}, y_{1})* obtained from

_{}

the quadratic equation is

_{},

which is one of a hyperbola. Graphically, the following diagram represents the curve:

The above two examples are particular cases of the following theorem that gives a general analysis of the quadratic forms in the plane.

** Theorem **Consider a quadratic form

_{}

in the plane.

1.
There is a counterclockwise rotation of the coordinate system
about the origin such that, in the new coordinate system, *f(x,y)* has no
cross term *xy.*

2.
The graph of *f(x,y)=D *is an ellipse if *C ^{2}-4AB*<0
and a hyperbola if

In the first example, *C ^{2}-4AB=1-4(1)(1)=-3<0*.
This is why the curve was an ellipse, and in the second