Case III: Repeated real eigenvalues.
 1 −2 2 


Example(5): Find the general solution of Y ' = −2 1 2 Y .


 2 2 1


Note: In the above example (5) we had repeated eigenvalues, and we can find two
linearly independent corresponding eigenvectors. But in general, this is not
always true. For this, consider the following example.
 0 1
Y .
−
9
6


Example(6): Consider the system Y ' = 
Note: From this example, we can see that there is only one linearly independent
eigenvector associated with the double eigenvalue.
The problem in this example is to find a second linearly independent
solution. So, how can we find this?
More generally, suppose that matrix A is an n  n matrix and it has
only k<n linearly independent eigenvectors. Then, the system Y ' = AY has
only k linearly independent solutions of the form X e
t
. Now our problem
is to find this additional n-k linearly independent solutions.
discussing this, we need to know the following.
Before
Algebraic and Geometric Multiplicity
Let A be an n  n matrix and 1 , 2 ,. . . , k be the distinct roots of A −  I = 0.
Then there exists positive integers m1 , m2 ,. . . , mk such that
A −  I = (−1) n ( − 1 ) m1 ( − 2 ) m2 . . . ( − k ) mk
and
m1 + m2 + . . . + mk = n . m j the number of times it is repeated as a root of the
characteristic equation is called the algebraic multiplicity(or simply
multiplicity) of the eigenvalue  j . We denote it as m( j ) or m j .
The number of linearly independent eigenvectors of A associated with  j is called
the geometric multiplicity of the eigenvalue  j and is denoted by
 ( j ) or  j .That is the dimension of the eigenspace corresponding to  j .
Theorem: For every eigenvalue  j , we have
 ( j )  m( j ).
Proof: Omit
Example: (1)
Consider the characteristic equation of a matrix A as,
( − 2)( + 3) 2 = 0 , then find the algebraic multiplicity of the eigenvalues.
(2)
If we have the characteristic equation of matrix A as
 ( − 1) ( + 1) = 0 , then find the algebraic multiplicity of the eigenvalues.
2
2
Definition: We call an eigenvalue  j of multiplicity m j complete if
 ( j ) = m( j ). Otherwise  j is called a defective or incomplete eigenvalue.
i.e.  ( j )  m( j ).
Remark: If the two multiplicities are not equal, then the eigenvalue is defective.
If A is a symmetric matrix, then all eigenvalues are complete.
Exercise: Find algebraic and geometric multiplicity of the eigenvalues of the
following coefficient matrices. Also, solve the following systems.
2 1
Y
'
=
(1)
 0 2 Y


(2)
 2 0
Y'=
Y
 0 2
(3)
1 0 0
Y ' =  1 1 0  Y
 2 −1 2 

