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Lecture IV
Eigenvalues, Eigenvectors, and Norms
I. Eigenvalues and Eigenvectors
Just to make sure that you haven’t picked up any bad habits, the determinant of any n * n matrix can be derived by expanding down any column or across any row of the matrix:
a a a a a a a a a a a a a a a a
a
a a a a a a a a a
a
a a a a a a a a a
a
a a a a a a a a a
a
a a a a a a a a a
11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44
1 1 11
22 23 24 32 33 34 42 43 44
2 1 21
12 13 14 23 33 34 42 43 44
3 1 31
12 13 14 22 23 24 42 43 44
4 1 41
12 13 14 22 23 23 32 33 34
where
a a a a a a a a a
a
a a a a a^
a a a a a^
a a a a
22 23 24 32 33 34 42 43 44
22
33 34 43 44 32
23 24 42 43 42
23 24 33 34
The eigenvalues of a matrix A are then defined as the solutions to the equation
A − λ I = 0
where I is the identity matrix. One example of the use of eigenvalues is from differential equations:
d v d t v w v at t d w d t
v w w at t
Another way to write this problem is
u t v t w t ( ) u A
^
^
^
^
Professor Charles B. Moss
Thus,
d u d t = A u t ( ) , u ( ) 0 = u 0.
We are looking for solutions of the form
u t e y w t e z
t t
λ λ.
Substituting this solution into the system of differential equations, we have
4 5 2 3
t t t t t t
e y e y e z e z e y e z
λ λ λ λ λ λ
(^) λ (^) − = λ^ −
next, we divide through by e λ t^ yielding
y z
y z y z
^
^
or
4 5 2 3
y y z z x A x
λ (^) = − λ =
which is the definition of the eigenvalues, λ, and the eigenvectors, x ,
A x x A I x
where the eigenvalues of x lie in the null space of the matrix
A − I =
^
More intuitively, we know that the matrix
4 5 2 3
Professor Charles B. Moss
f ( ) x − f ( x *) = ∇ x (^) f ( ) x dx + 12 dx ∇ (^) xx^2 f ( ) x dx '≤ 0
which implies for optimality that
∇ (^) x f ( ) x (^) p →0.
The matrix norm is analogously defined with the additional property that 4.) AB ≤ A B.
Another value of the norm is a discussion of the stability of the matrix. For example, what are the properties of the solution of Ax = b if we perturb the x vector?
A x ( x ) (^) ( b b ) x x A b A b
+ = −^ + −
δ δ δ^1 1 δ^
Noting that x = A -1 b we subtract x and take the norm of each side:
δ x = A −^1 δ b.
Given that b ≤ A x by the original equation, we have
1 x
A
b
Multiplying both side by the inequality involving the perturbation, δ, we get
δ x δ
x
A A b b
≤ −^1.
This number A A −^1 is the condition number of A denoted cond( A ). Since
1 = I ≤ A A −^1
cond( A ) ≥ 1. If the condition number of A is large, the matrix is ill-conditions. If the condition number is small, the matrix is well conditioned.
III. Matrix Factorizations The matrix factorization A = LU involves reducing a matrix into a lower diagonal matrix L and an upper diagonal matrix U. The LU factorization is computed by a set of elementary row operations. Taking the matrix
Professor Charles B. Moss
A =
the first elementary matrix is
E 1
The resulting matrix is
A 1
the second factorization is then
E 2
which yields the upper triangular matrix
A 2 U
The lower triangular matrix is then derived as
L = E E = −
− − 1 1 2 1
The usefulness of the decomposition is twofold. First, the LU factorization is a quick way to solve large systems of equations. First, note that the solution of the system of linear equations Ax = b for b =(1,5,3)’ is x =(-5/2,4,1)’. If we start with the augmented system of equations:
Professor Charles B. Moss
with the upper triangular matrix as
U = −
Noting that the diagonal of the L matrix is 1 and the diagonal of the U matrix are greater than zero, suppose that we constructed the matrix
D −^ =
1
which is one over the diagonal elements of the U matrix. Multiplying D -1^ U yields L ’.
