Study Guide for Quiz 8
1. Eigenvalues
Given a 2 × 2 matrix A, its eigenvalues λ and µ satisfy the equations:
λ + µ = trace(A)
and
λµ = det(A).
You can use this to find the eigenvalues of any 2 × 2 matrix.
2. The Matrix for a Quadratic Form
Any quadratic form f (x, y) = ax2 + bxy + cy 2 on R2 can be written using a matrix:
#" #
"
a b/2 x
.
f (x, y) = x y
y
b/2 c
Similarly, a quadratic form f (x, y, z) = ax2 + by 2 + cz 2 + pxy + qxz + ryz can be written using
a matrix:

 
a p/2 q/2 x

 
f (x, y) = x y z p/2 b r/2 y  .
q/2 r/2 c
z
3. Classifying Quadratic Forms
You can use the eigenvalues of the matrix to determine the shape of the graph:
• If both eigenvalues are positive, the graph is an elliptic paraboloid that opens up.
• If both eigenvalues are negative, the graph is an elliptic paraboloid that opens down.
• If one eigenvalue is positive and the other is negative, the graph is a saddle surface.
In the first two cases, the level curves for f are ellipses, while in the third case they are
hyperbolas.
4. Definite Forms
A quadratic form is positive definite if all of its eigenvalues are positive, and negative
definite if all of its eigenvalues are negative. According to Sylvester’s criterion, a quadratic
form

 
a
r
s
x



y
f (x, y, z) = x y z r b t
s t c
z
is positive definite if and only if
a>0
and
a r
r b > 0
and
a r s
r b t > 0.
s t c
You can also use this to check whether a quadratic form f : R3 → R is negative definite: just
check whether the negation −f is positive definite.
5. Critical Points
A critical point for a function f : R2 → R is a point for a ∈ R2 for which
∂f
(a) = 0
∂x
∂f
(a) = 0.
∂y
and
More generally, a critical point for a function f : Rn → R is a point for a ∈ Rn for which
∂f
(a) = 0,
∂x1
∂f
(a) = 0,
∂x2
...,
and
∂f
(a) = 0.
∂xn
To find the critical points for a function f , set all of the partial derivatives equal to zero and
then solve the resulting system of equations.
6. The Hessian Matrix
The Hessian of a function f : R2 → R is the following 2 × 2 matrix:
 2

∂ f
∂ 2f
 ∂x2 ∂x∂y 


Hf = 

 ∂ 2f
∂ 2f 
∂x∂y ∂y 2
Similarly, the Hessian of a function f : R3 → R is the following 3 × 3 matrix:

 2
∂ f
∂ 2f
∂ 2f
 ∂x2 ∂x∂y ∂x∂z 



 2
2
2

 ∂ f
∂
f
∂
f

Hf = 
 ∂x∂y ∂y 2 ∂y∂z 




2
2
 ∂ 2f
∂ f
∂ f 
∂x∂z ∂y∂z ∂z 2
The Hessian is a multivariable analogue of the second derivative of a function. We use the
notation Hf (a) to denote the value of the Hessian matrix at a point a.
7. The Second Derivative Test
Let f : Rn → R be a differentiable function, and let a be a critical point of f . Then:
(a) If the Hessian Hf (a) is positive definite, then f has a local minimum at a.
(b) If the Hessian Hf (a) is negative definite, then f has a local maximum at a.
(c) If the Hessian Hf (a) has both negative and positive eigenvalues, then f has neither a
local minimum nor a local maximum at a.
8. Multivariable Taylor Series
The Taylor series for a function f : R2 → R has the form
∞ X
∞
X
m=0 n=0
cm,n xm y n
where
cm,n =
1 ∂ m+n f
(0).
m! n! ∂xm ∂y n
The second-order Taylor polynomial p2 (x, y) is the sum of the constant, linear, and
quadratic terms of this Taylor series:
p2 (x, y) = f (0) +
∂f
∂f
1 ∂ 2f
1 ∂ 2f
∂ 2f
2
(0) x +
(0) y +
(0)
xy
+
(0)
x
+
(0) y 2
∂x
∂y
2 ∂x2
∂x∂y
2 ∂y 2
This can also be written
p2 (x) = f (0) + Df (0) x +
1 T
x Hf (0) x.
2