A First Course in Linear Algebra, 2017a

432 Spectral Theory
Exercise 7.4.25 A quadratic form in three variables is an expression of the form a 1 x 2 + a 2 y 2 + a 3 z 2 +
a 4 xy + a 5 xz + a 6 yz. Show that every such quadratic form may be written as
⎡ ⎤
[ ]
x
x y z A ⎣ y ⎦
z
where A is a symmetric matrix.
Exercise 7.4.26 Given a quadratic form in three variables, x,y, and z, show there exists an orthogonal
matrix U and variables x ′ ,y ′ ,z ′ such that
⎡ ⎤ ⎡
x x ′ ⎤
⎣ y ⎦ = U ⎣ y ′ ⎦
z z ′
with the property that in terms of the new variables, the quadratic form is
λ 1
(
x
′ ) 2 + λ2
(
y
′ ) 2 + λ3
(
z
′ ) 2
where the numbers, λ 1 ,λ 2 , and λ 3 are the eigenvalues of the matrix A in Problem 7.4.25.
Exercise 7.4.27 Consider the quadratic form q given by q = 3x 2 1 − 12x 1x 2 − 2x 2 2 .
(a) Write q in the form⃗x T A⃗x for an appropriate symmetric matrix A.
(b) Use a change of variables to rewrite q to eliminate the x 1 x 2 term.
Exercise 7.4.28 Consider the quadratic form q given by q = −2x 2 1 + 2x 1x 2 − 2x 2 2 .
(a) Write q in the form⃗x T A⃗x for an appropriate symmetric matrix A.
(b) Use a change of variables to rewrite q to eliminate the x 1 x 2 term.
Exercise 7.4.29 Consider the quadratic form q given by q = 7x 2 1 + 6x 1x 2 − x 2 2 .
(a) Write q in the form⃗x T A⃗x for an appropriate symmetric matrix A.
(b) Use a change of variables to rewrite q to eliminate the x 1 x 2 term.