A First Course in Linear Algebra, 2017a

424 Spectral Theory
= (U⃗y) T A(U⃗y)
= ⃗y T (U T AU )⃗y
= ⃗y T D⃗y
The following procedure details the steps for the change of variables given in the above theorem.
Procedure 7.93: Diagonalizing a Quadratic Form
Let q be a quadratic form in the variables x 1 ,···,x n given by
q = a 11 x 2 1 + a 22 x 2 2 + ···+ a nn x 2 n + a 12 x 1 x 2 + ···
Then, q can be written as q = d 11 y 2 1 + ···+ d nny 2 n as follows:
1. Write q =⃗x T A⃗x for a symmetric matrix A.
2. Orthogonally diagonalize A to be written as U T AU = D for an orthogonal matrix U and
diagonal matrix D.
⎡ ⎤
y 1
y 2
3. Write⃗y = ⎢ .
⎣ ..
⎥. Then,⃗x = U⃗y.
⎦
y n
4. The quadratic form q will now be given by
q = d 11 y 2 1 + ···+ d nny 2 n =⃗yT D⃗y
where D = [ d ij
]
is the diagonal matrix found by orthogonally diagonalizing A.
Consider the following example.
Example 7.94: Choosing New Axes to Simplify a Quadratic Form
Consider the following level curve
6x 2 1 + 4x 1x 2 + 3x 2 2 = 7
shown in the following graph.
x 2
x 1
Use a change of variables to choose new axes such that the ellipse is oriented parallel to the new
coordinate axes. In other words, use a change of variables to rewrite q to eliminate the x 1 x 2 term.