6.17. Summary
Theorem 6.1.
m T    tij   diag  1 ,
, n   T  vk   k vk
(6.1)
6.2.1. Definition: Eigenvalue and Eigenvector.
Eigen-equation
T  x   x
(6.2)
 is called an eigenvalue of T and x is an eigenvector belonging to .
Theorem 6.2.
The eigenvectors u1 ,
, uk belonging to a set of distinct eigenvalues 1 ,
, k of T
are independent.
Theorem 6.3.
T : V  V has at most n  dimV distinct eigenvalues.
eigenvalues 1 , , n , then
m T   diag  1,
If T has exactly n distinct
, n 
if we use the corresponding eigenvectors as basis.
Theorem 6.4.
Let V be a finite- dimensional linear space with complex scalars. Then every linear
transformation T : V  V has an eigenvalue.
Theorem 6.5.
Triangularization Theorem.
Given any linear transformation T : V  V , where V has complex scalars and
dimV  n . There is a basis for V relative to which m T  is upper triangular.
Furthermore, each diagonal element of m T  is an eigenvalue of T.
6.7.1. Secular Equation
det   I  A  0
(6.10)
6.7.2. Theorem 6.6.
Let A   aij  be an n  n matrix, then
f     det   I  A
(6.10a)
is a polynomial of degree n in . Furthermore, the coefficient of  n is 1 and the
constant term is f  0  det   A     det A .
n
6.7.3. Characteristic Polynomials
Definition:
Characteristic Polynomial
The determinant
f     det   I  A
is called the characteristic polynomial of A.
Theorem 6.7.
The set of eigenvalues of T : V  V consists of all the roots of the characteristic
polynomial of m T  that lie in the scalar field F of V.
6.9. Product And Sum
n
det A   i
i 1
n
trA   aii
i 1
6.11.1. Theorem 6.8.
If two n  n matrices A and B represent the same linear transformation T, then there is
a nonsingular matrix C such that
B  C 1 AC
Furthermore, if A and B are the representations relative to the bases V   v1,
and U   u1,
, vn 
, un  , respectively, then U  VC .
6.11.2. Theorem 6.9.
Let A and B be two n  n matrices related by B  C 1 AC , where C is n  n and
nonsingular. Then A and B represent the same linear transformation.
6.11.3.
Definition
Two n  n matrices A and B are similar if there exists a nonsingular matrix C such
that B  C 1 AC .
Theorem 6.10.
Two n  n matrices are similar iff they represent the same linear transformation.
Theorem 6.11.
Similar matrices have the same characteristic polynomial and hence the same
eigenvalues.
Theorem 6.12.
Let W be an n-D linear space over the scalar field F and T : W  W is a linear
transformation. If the characteristic polynomial of T has n distinct roots 1 , , n
in F, then
(a) The corresponding eigenvectors u1 ,
(b) m T     diag  1,
, un form a basis for W.
, n  relative to the ordered basis U   u1,
(c) If m T   A relative to another ordered basis V   v1,
, un  .
, vn  , then
  C 1 AC , where C is the transition matrix given by U  VC .
6.13.1. Theorem 6.13:
Operators
Cayley-Hamilton Theorem For Linear
Given a characteristic polynomial for T : V  V ,
n
f         i 
(6.24)
i 1
Then
n
f T    T  i I   O
i 1
where O is the zero operator, i.e.,
f T  x   O
Corollary 6.14:
 x V
(6.25)
Cayley-Hamilton Theorem For Matrices
Let A be an n  n matrix with characteristic polynomial
n
f     det   I  A   ck  k
k 0
Then
n
f  A   ck Ak  O
k 0
where O is the zero operator.
6.15.1.
Definition:
Jordan Block
A Jordan block is a upper triangular n  n matrix J   jik  such that jik  0
except
jkk  
and
for k  1,
jk ,k 1  1
,n
where  is a scalar. Some examples are

 1 0 
0  1


0 0 


 1 
0 


 1 0 0 
0  1 0


0 0  1


0 0 0 
Theorem 6.15.
A Jordan block J   jik  of order n has exactly one eigenvalue jkk and one
independent eigenvector 1,0,
,0 .
6.15.2. Theorem 6.16: Jordan Normal Form
Let V be an n-D linear space with complex scalars. Let T : V  V be a linear
transformation. Then there exists a basis for V relative to which
m T   diag  J1,
, Jm 
where each J k is a Jordan block.
[ Block diagonal ]