Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

6 CHAPTER 1. INTRODUCTION
Example Problem 1.12. Find the eigenvalues of
[ 1 1
4 −2
]
Solution: The eigenvalues are roots of
[ ]
1 − λ 1
0 = det
= (1 − λ) (−2 − λ) − 4 = λ 2 + λ − 6.
4 −2 − λ
This equation has roots λ 1 = −3, λ 2 = 2.
⊣
Example Problem 1.13. Find the eigenvalues of A 2 .
Solution: Let λ be an eigenvalue of A, with corresponding eigenvector x. Then
A 2 x = A (Ax) = A (λx) = λAx = λ 2 x.
The eigenvalues of a matrix tell us, roughly, how the linear transform scales a given matrix; the
eigenvectors tell us which directions are “purely scaled.” This will make more sense when we talk
about norms of vectors and matrices.
1.3.1 Matrix Norms
We define what we mean by a “norm” on the space R n :
Definition 1.14. A function ‖·‖ from R n to R + is called a norm if
1. It obeys the triangle inequality: ‖x + y‖ ≤ ‖x‖ + ‖y‖ .
2. It scales positively: ‖αx‖ = |α| ‖x‖ , for real α.
3. It is positive: ‖x‖ ≥ 0, with equality only holding when x is the zero vector.
We usually use the following norm, which should be familiar to you.
Definition 1.15. For vector x ∈ R n , its “two-norm” is defined
‖x‖ 2 =
( n∑
i=1
x 2 i
) 1
2
=
( ) 1
x ⊤ 2
x .
⊣
You should verify that this is indeed a norm in the sense outlined above.
Given a vector norm, we can define the matrix norm “subordinate” to it, as follows:
Definition 1.16. Given a norm ‖·‖ on R n , we define the subordinate matrix norm on R n×n by
‖A‖ = max
x≠0
‖Ax‖
‖x‖ .
We will use the subordinate two-norm for matrices. From the definition of the subordinate
norm as a max, we conclude that if x is a nonzero vector then
‖Ax‖ 2
‖x‖ 2
≤ ‖A‖ 2
thus,
‖Ax‖ 2
≤ ‖A‖ 2
‖x‖ 2
.