MATH 304
Linear Algebra
Lecture 32:
Bases of eigenvectors.
Diagonalization.
Eigenvalues and eigenvectors of an operator
Definition. Let V be a vector space and L : V → V
be a linear operator. A number λ is called an
eigenvalue of the operator L if L(v) = λv for a
nonzero vector v ∈ V . The vector v is called an
eigenvector of L associated with the eigenvalue λ.
(If V is a functional space then eigenvectors are also
called eigenfunctions.)
If V = Rn then the linear operator L is given by
L(x) = Ax, where A is an n×n matrix.
In this case, eigenvalues and eigenvectors of the
operator L are precisely eigenvalues and
eigenvectors of the matrix A.
Characteristic polynomial of an operator
Let L be a linear operator on a finite-dimensional
vector space V . Let u1 , u2 , . . . , un be a basis for V .
Let A be the matrix of L with respect to this basis.
Definition. The characteristic polynomial
p(λ) = det(A − λI ) of the matrix A is called the
characteristic polynomial of the operator L.
Then eigenvalues of L are roots of its characteristic
polynomial.
Theorem. The characteristic polynomial of the
operator L is well defined. That is, it does not
depend on the choice of a basis.
Theorem. The characteristic polynomial of the
operator L is well defined. That is, it does not
depend on the choice of a basis.
Proof: Let B be the matrix of L with respect to a
different basis v1 , v2 , . . . , vn . Then A = UBU −1 ,
where U is the transition matrix from the basis
v1 , . . . , vn to u1 , . . . , un . We have to show that
det(A − λI ) = det(B − λI ) for all λ ∈ R. We
obtain
det(A − λI ) = det(UBU −1 − λI )
= det UBU −1 − U(λI )U −1 = det U(B − λI )U −1
= det(U) det(B − λI ) det(U −1 ) = det(B − λI ).
Let V be a vector space and L : V → V be a linear
operator.
Proposition 1 If v ∈ V is an eigenvector of the
operator L then the associated eigenvalue is unique.
Proof: Suppose that L(v) = λ1 v and L(v) = λ2 v. Then
λ1 v = λ2 v =⇒ (λ1 − λ2 )v = 0 =⇒ λ1 − λ2 = 0 =⇒ λ1 = λ2 .
Proposition 2 Suppose v1 and v2 are eigenvectors
of L associated with different eigenvalues λ1 and λ2 .
Then v1 and v2 are linearly independent.
Proof: Since L(v1 ) = λ1 v1 , we have
L(tv1 ) = tL(v1 ) = t(λ1 v1 ) = λ1 (tv1 ) for any scalar t. Since
λ2 6= λ1 , it follows that v2 6= tv1 . That is, v2 is not a scalar
multiple of v1 . Similarly, v1 is not a scalar multiple of v2 .
Let L : V → V be a linear operator.
Proposition 3 If v1 , v2 , and v3 are eigenvectors of
L associated with distinct eigenvalues λ1 , λ2 , and
λ3 , then they are linearly independent.
Proof: Suppose that t1 v1 + t2 v2 + t3 v3 = 0 for some
t1 , t2 , t3 ∈ R. Then
L(t1 v1 + t2 v2 + t3 v3 ) = 0,
t1 L(v1 ) + t2 L(v2 ) + t3 L(v3 ) = 0,
t1 λ1 v1 + t2 λ2 v2 + t3 λ3 v3 = 0.
It follows that
t1 λ1 v1 + t2 λ2 v2 + t3 λ3 v3 − λ3 (t1 v1 + t2 v2 + t3 v3 ) = 0
=⇒ t1 (λ1 − λ3 )v1 + t2 (λ2 − λ3 )v2 = 0.
By the above, v1 and v2 are linearly independent.
Hence t1 (λ1 − λ3 ) = t2 (λ2 − λ3 ) = 0
=⇒ t1 = t2 = 0 =⇒ t3 = 0.
Theorem If v1 , v2 , . . . , vk are eigenvectors of a
linear operator L associated with distinct
eigenvalues λ1 , λ2 , . . . , λk , then v1 , v2 , . . . , vk are
linearly independent.
Corollary 1 If λ1 , λ2 , . . . , λk are distinct real
numbers, then the functions e λ1 x , e λ2 x , . . . , e λk x are
linearly independent.
Proof: Consider a linear operator
D : C ∞ (R) → C ∞ (R) given by Df = f ′ .
Then e λ1 x , . . . , e λk x are eigenfunctions of D
associated with distinct eigenvalues λ1 , . . . , λk .
Corollary 2 Let A be an n×n matrix such that
the characteristic equation det(A − λI ) = 0 has n
distinct real roots. Then Rn has a basis consisting
of eigenvectors of A.
Proof: Let λ1 , λ2 , . . . , λn be distinct real roots of the
characteristic equation. Any λi is an eigenvalue of A, hence
there is an associated eigenvector vi . By the theorem, vectors
v1 , v2 , . . . , vn are linearly independent. Therefore they form a
basis for Rn .
Corollary 3 Let λ1 , λ2 , . . . , λk be distinct
eigenvalues of a linear operator L. For any
1 ≤ i ≤ k let Si be a basis for the eigenspace
associated with the eigenvalue λi . Then the union
S1 ∪ S2 ∪ · · · ∪ Sk is a linearly independent set.
Diagonalization
Let L be a linear operator on a finite-dimensional vector space
V . Then the following conditions are equivalent:
• the matrix of L with respect to some basis is diagonal;
• there exists a basis for V formed by eigenvectors of L.
The operator L is diagonalizable if it satisfies these
conditions.
Let A be an n×n matrix. Then the following conditions are
equivalent:
• A is the matrix of a diagonalizable operator;
• A is similar to a diagonal matrix, i.e., it is represented as
A = UBU −1 , where the matrix B is diagonal;
• there exists a basis for Rn formed by eigenvectors of A.
The matrix A is diagonalizable if it satisfies these conditions.
Otherwise A is called defective.
Example. A =
2 1
.
1 2
• The matrix A has two eigenvalues: 1 and 3.
• The eigenspace of A associated with the
eigenvalue 1 is the line spanned by v1 = (−1, 1).
• The eigenspace of A associated with the
eigenvalue 3 is the line spanned by v2 = (1, 1).
• Eigenvectors v1 and v2 form a basis for R2 .
Thus the matrix A is diagonalizable. Namely,
A = UBU −1 , where
1 0
−1 1
B=
,
U=
.
0 3
1 1
Example. A
•
•
v1
•
v2
•


1 1 −1
= 1 1 1 .
0 0
2
The matrix A has two eigenvalues: 0 and 2.
The eigenspace corresponding to 0 is spanned by
= (−1, 1, 0).
The eigenspace corresponding to 2 is spanned by
= (1, 1, 0) and v3 = (−1, 0, 1).
Eigenvectors v1 , v2 , v3 form a basis for R3 .
Thus the matrix A is diagonalizable. Namely,
A = UBU −1 , where
B=

0 0 0
 0 2 0 ,
0 0 2

U=

−1 1 −1
 1 1
0 .
0 0
1

Problem. Diagonalize the matrix A =
4 3
.
0 1
We need to find a diagonal matrix B and an
invertible matrix U such that A = UBU −1 .
Suppose that v1 = (x1 , y1 ), v2 = (x2 , y2 ) is a basis
for R2 formed by eigenvectors of A, i.e., Avi = λi vi
for some λi ∈ R. Then we can take
x1 x2
λ1 0
.
,
U=
B=
y1 y2
0 λ2
Note that U is the transition matrix from the basis
v1 , v2 to the standard basis.
4 3
Problem. Diagonalize the matrix A =
.
0 1
4−λ
3
= 0.
Characteristic equation of A: 0
1−λ
(4 − λ)(1 − λ) = 0 =⇒ λ1 = 4, λ2 = 1.
Associated eigenvectors: v1 = (1, 0), v2 = (−1, 1).
Thus A = UBU −1 , where
4 0
1 −1
B=
,
U=
.
0 1
0 1
Problem. Let A =
4 3
. Find A5 .
0 1
We know that A = UBU −1 , where
4 0
1 −1
B=
,
U=
.
0 1
0 1
Then A5 = UBU −1 UBU −1 UBU −1 UBU −1 UBU −1
1
−1
1024
0
1
1
= UB 5 U −1 =
0 1
0 1
0 1
1024 −1
1 1
1024 1023
=
=
.
0
1
0 1
0
1
Problem. Let A =
4 3
. Find a matrix C
0 1
such that C 2 = A.
We know that A = UBU −1 , where
1 −1
4 0
.
,
U=
B=
0
1
0 1
Suppose that D 2 = B for some matrix D. Let C = UDU −1 .
Then C 2 = UDU −1 UDU −1 = UD 2 U −1 = UBU −1 = A.
√
2 0
4 √0
We can take D =
.
=
0 1
1
0
2 1
1 1
2 0
1 −1
.
=
Then C =
0 1
0 1
0 1
0
1
There are two obstructions to existence of a basis
consisting of eigenvectors. They are illustrated by
the following examples.
1 1
Example 1. A =
.
0 1
det(A − λI ) = (λ − 1)2 . Hence λ = 1 is the only
eigenvalue. The associated eigenspace is the line
t(1, 0).
0 −1
Example 2. A =
.
1 0
det(A − λI ) = λ2 + 1.
=⇒ no real eigenvalues or eigenvectors
(However there are complex eigenvalues/eigenvectors.)