A First Course in Linear Algebra, 2017a

7.4. Orthogonality 421
Usually it won’t work out so well but you can still find what is desired. Thus, once you have obtained
approximate eigenvalues using the QR algorithm, you can find the eigenvalue more exactly along with an
eigenvector associated with it by using the shifted inverse power method.
♠
7.4.5 Quadratic Forms
One of the applications of orthogonal diagonalization is that of quadratic forms and graphs of level curves
of a quadratic form. This section has to do with rotation of axes so that with respect to the new axes,
the graph of the level curve of a quadratic form is oriented parallel to the coordinate axes. This makes
it much easier to understand. For example, we all know that x 2 1 + x2 2
= 1 represents the equation in two
variables whose graph in R 2 is a circle of radius 1. But how do we know what the graph of the equation
5x 2 1 + 4x 1x 2 + 3x 2 2 = 1 represents?
We first formally define what is meant by a quadratic form. In this section we will work with only real
quadratic forms, which means that the coefficients will all be real numbers.
Definition 7.90: Quadratic Form
A quadratic form is a polynomial of degree two in n variables x 1 ,x 2 ,···,x n , written as a linear
combination of x 2 i terms and x i x j terms.
⎡ ⎤
x 1
Consider the quadratic form q = a 11 x 2 1 + a 22x 2 2 + ···+ a nnx 2 n + a x 2 12x 1 x 2 + ···. We can write⃗x = ⎢ ⎥
⎣ . ⎦
x n
as the vector whose entries are the variables contained in the quadratic form.
⎡
⎤
a 11 a 12 ··· a 1n
a 21 a 22 ··· a 2n
Similarly, let A = ⎢
⎥
⎣ . . . ⎦ be the matrix whose entries are the coefficients of x2 i and
a n1 a n2 ··· a nn
x i x j from q. Note that the matrix A is not unique, and we will consider this further in the example below.
Using this matrix A, the quadratic form can be written as q =⃗x T A⃗x.
q
= ⃗x T A⃗x
⎡
= [ ]
x 1 x 2 ··· x n ⎢
⎣
⎡
= [ ]
x 1 x 2 ··· x n ⎢
⎣
⎡ ⎤
x 1
x 2 ⎢ ⎥
⎣ . ⎦
x n
⎤
a 11 x 1 + a 21 x 2 + ···+ a n1 x n
a 12 x 1 + a 22 x 2 + ···+ a n2 x n
⎥
.
⎦
a 1n x 1 + a 2n x 2 + ···+ a nn x n
⎤
a 11 a 12 ··· a 1n
a 21 a 22 ··· a 2n
⎥
. . . ⎦
a n1 a n2 ··· a nn