2026 Prof.Jiang@ECE NYU 299
Lecture VII
• The Jordan Canonical Form
• Examples and Applications
2026 Prof.Jiang@ECE NYU 300
Review of Canonical Forms
 1
If is an matrix with eigenvalues,
then there exists a nonsingular matrix s.t.
diag .
If is (possibly having repeated
distinct
He eigenvalues),
a unitary ma
rmitian
t
i
A n n
P
P AP
A

 
 


 *
rix such that
diag . i
U
U AU  
2026 Prof.Jiang@ECE NYU 301
A Motivating Example
As stated previously, not every matrix can be
transformed into a canonical diagonal form.
Fo
can
r e
not
xample,
1
, 0
0 1
be transformed into a diagonal matrix.
b
A b    
2026 Prof.Jiang@ECE NYU 302
Jordan Canonical Form
   
1 1
1
(Jordan):
Let be an matrix whose eigenvalues
are , , with multiplicities , , :
det
Then, is transformable
dif
int
ferent
nonsin
o a
Th
Jordan canonical form.
i.e.,
eorem

s s
i
i
s m
i
A n n
m m
I A
A


 
    


 
 1
such that
blo
gula
ckdiag
r
i
P
P AP J   
2026 Prof.Jiang@ECE NYU 303
 1
(Jordan), :
blockdiag
0
1

1 0
0 0
Theorem con
1
t d'
0
0
0 0
i
i
i
i
i
i
P AP J
where
  
          

 
 
    

 
Jordan
Block
2026 Prof.Jiang@ECE NYU 304
Comments
, is used as .
Different Jordan blocks, say , may be
associated with the eigenvalues.
The total number of Jordan blocks: .
Jordan
e
for
s
m
am
T
i j
In some texts J
s s s n

  
   
2026 Prof.Jiang@ECE NYU 305
Illustration via 3x3 matrices
1
1 1
1 1 2 1
1 1
1
3 1
1
If a 3 3 matrix has an eigenvalue
of multiplicity three, then it may be reduced
into one of the following Jordan forms:
0 0 0 0
0 0 , 0 0 ,
0 0 0 1
0 0
1 0
0 1
A
J J
J
 
                    
      
.

2026 Prof.Jiang@ECE NYU 306
Remark 1
 The distinct Jordan forms , , , are similar
to
not
each other.
i kJ J i k
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Remark 2
When each Jordan block ( ) in the Jordan
form is one-dimensional (i.e. 1) and ,
the Jordan matrix becomes diagonal.
i i
iJ n s n
J

 
2026 Prof.Jiang@ECE NYU 308
Application to Matrix Analysis of
Differential Equations
 
1
1
1
Given a set of 1st-order differential equations
( ) ( ), (0) ,
applying the transformation yields:
( ) ( ) : ( ).
( ) ( ), , i
n
mi i i
i
s
x t Ax t x
y P x
y t P AP y t Jy t
y
t y t y y
y
y


 

 
       

 

   .

2026 Prof.Jiang@ECE NYU 309
Comment
So, with the help of Jordan canonical form,
solving differential equations can be
reduced down to solving lower-order
(disjoint!) differential equations.
(see a forthcoming lecture.)
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Principal Vectors
In order to develop a constructive method for
resulting in Jordan form, let's introduce the notion of
, or generalized eigenvector,
which is a generalization of eigenvector.
principal vector
P
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Principal Vectors
 
A (possibly zero) vector is a
0 belonging to the eigenvalue if
0,
princ
for w
ipal ve
hich is the smallest non-negativ
ctor
of gra
e inte er
e
g
d
.
i
g
i
p
g
I A p
g
 
  
2026 Prof.Jiang@ECE NYU 312
Examples
• The vector p = 0 is the principal vector of
grade 0.
• The (nonzero) eigenvectors are the
principal vectors of grade 1.
2026 Prof.Jiang@ECE NYU 313
Motivating Question
 1
In case of transformation to diagonal
canonical form, i.e., diag ,
the columns of are linearly independent
eigenvectors.
What about the matrix in Jordan form?
How to construct from principal
iP AP
P
P
P
  
vectors?
2026 Prof.Jiang@ECE NYU 314
Linear Spaces
    
 
     0 1 2
Define the linear space composed of all principal
vectors of grade belonging to :
| 0
. ., the null space of .
Clearly,

g
g
g
i
i
i i
i i i
g
P p I A p
i e I A
P P P
 
    
 
     
2026 Prof.Jiang@ECE NYU 315
An Interesting Result
1
1
1 2
Let be an matrix with the distinct
eigenvalues , , , 1 , with
multiplicities , , .
Then, vector can be written as

where is a uniquely defined princi
every
p
s
s
n
s
i
A n n
s n
m m
x
x p p p
p

   

   




al vector
associated with of grade .i im 
2026 Prof.Jiang@ECE NYU 316
Comment 1
1
1 1
1
A special, but interesting, case is when there
are linearly independent eigenvectors, say,
, , . In this case, scalars s.t.
: .
n
n n
n
i
n
c c
x c c p p

       

 
2026 Prof.Jiang@ECE NYU 317
Comment 2
Its proof relies upon the well-known Cayley-
Hamilton theorem; see any standard
matrix or linear algebra textbook.
2026 Prof.Jiang@ECE NYU 318
Example
1 0 0
Consider the matrix 7 1 0 .
0 0 2
Compute its eigenvalues and the associated eigenvectors.
* Can each column be written as a linear combination
of eigenvectors?
* Show that each column c
A
      

an be written
as a unique representation of principal vectors.
2026 Prof.Jiang@ECE NYU 319
Answer
   
 
 
    
1 1 2 2
1
2
1 1
2 1
1 2 , 2 1 .
eigenvectors of are of the form
col 0,1, 0 , any nonzero scalar.
eigenvectors of are of the form
col 0, 0,1 , any nonzero scalar.
| , , 0

m m
The
The
P p p col
      
 
 

 
     
    2 21 2 | 0, 0, .P p p col   
2026 Prof.Jiang@ECE NYU 320
Cayley-Hamilton Theorem Revisited
 
   
1
1 1
1
For any matrix ,
( )
is the characteristic polynomial
of , i.e.,
det .
n n
n n
n
n n i
i
i
n n A
A A A A I O
where
A
I A





      
 
         

2026 Prof.Jiang@ECE NYU 321
Example
2
2
7 4
Consider . Verify that
8 5
1) The characteristic polynomial ( ) is:
( ) 2 3.
0 0
2) ( ) 2 3 0 .
0 0
A
A
A
A
A A A I
 
   

     
  
       
2026 Prof.Jiang@ECE NYU 322
Another Proof
   
 
     
  1 0 1
Define the matrix of signed cofactors:
cof .
, using cof (det ) ,
.
In addition,

for constant matrices ' .
T
T
T n
n n
i
n n
C I A
Then M M M I
I A C I
C C C C
C s



   

     
      
2026 Prof.Jiang@ECE NYU 323
Proof (cont’d)
0
1 0 1
1 2 1
1
identification of the coefficients of equal powers
of gives




.
Multiplying the first eq. by ,
n n n
n n
n
By
C I
C AC I
C AC I
AC I
A
  



  
  
  

 
1the second by ,...,
then adding them up leads to: .
nA
and O A I

 
2026 Prof.Jiang@ECE NYU 324
Question
How to compute principal vectors for a
given matrix?
2026 Prof.Jiang@ECE NYU 325
A Motivating Example
1 2
1
1 2 1 2
2 2 2
0
Consider a 2 2 Jordan block .
1
Denote that transforms into .
, . So, we have , or
0

1
, so is an eigenvector;
(
J
P x x A J
Namely P AP J AP PJ
A x x x x
Ax x x






     
   
 
           
 
1 2 1) , so is a principal vector (of grade 2).A I x x x 
2026 Prof.Jiang@ECE NYU 326
Comment
 1 2Usually, , is called a for
this 2 2 matrix . In order words,
the JCF transformation matrix is composed
of a Jordan basis, o linearly independenr ta set of
eigenvecto
Jordan Bas
rs and pri cipa
is
n l
x x
A
P

vectors.
2026 Prof.Jiang@ECE NYU 327
General Procedure
 
 
 
1
1
2 1
22 2
2
Solve the characteristic equation
0.
For each independent , solve

where clearly
Step
linearl
solves 0.
Collect o y innly
1
th
:
ose z which dare
2 :
epe
A I z
z
A I z z
z A
Ste
z
p
I



 
 
 
1with the previously found eigenvectors .

ndent

z
2026 Prof.Jiang@ECE NYU 328
General Procedure
 
 
2
3 2
33 3
3
1 2
For each independent , solve

where clearly solves 0.
Collect on linearly independenly those z which are
with the previously found vectors
Ste
, .
Contin
p 3:
Ste uep :
t
4
z
A I z z
z A I z
z z


 
 
in this way till the total number of
independent eigenvectors and principal vectors equals
to the (algebraic) multiplicity of .


2026 Prof.Jiang@ECE NYU 329
General Procedure
1 2 1 1
1 2
1
Denote
, , , , , ,
, , , .
Therefore,
(associated with eigenvalue
Step 4 (co
).

nt

'

d :

)
m m m
m
x x x z z z
and
P x x x
P AP J 


      
   

 

eigenvector
2026 Prof.Jiang@ECE NYU 330
Comments
• Not any arbitrary choice of linearly independent
principal vectors would lead to a correct transformation
matrix P.
•
 
 
2
2 1
2 1
,
principal vectors are chosen according to

2,
:

linearly ind
.
ependentFor example at the
z
A I z z
but NOT
I A z z
Step


 
 
See (the 1960 book of Gantmacher, Vol.1, Chap. VI,
Section 8) for another general method of constructing
a transformation matrix.
2026 Prof.Jiang@ECE NYU 331
More on Jordan Basis
Without going into the full details in proving Jordan's
Theorem, let's illustrate the concept of Jordan basis
and its use in the canonical transformation.
Consider a principal vector of grade 4. Dev g n 
 
 
 
1
2 1
3 2
4 3
fine:
:
:


Jo
:
:
rdan Basis
x v
x A I x
x A I x
x A I x
       
2026 Prof.Jiang@ECE NYU 332
Jordan Basis (cont’d)
 1 1 2 3 4
Then, the 4 4 matrix can be transformed into
the Jordan canonical form:
0 0 0
1 0 0

0 1 0
0 0 1
,
, , , , .
A
J
That is
P AP J P x x x x

       
 
eigenvector
Principal vector
of grade 4
2026 Prof.Jiang@ECE NYU 333
Comment
 4 3 2 1
1
If we define , , , , then is transformed
into the Jordan canonical form , i.e.:
1 0 0
0 1 0
.
0 0 1
0 0 0
T
T
P x x x x A
J
P AP J

        

 
2026 Prof.Jiang@ECE NYU 334
A More Complex Case
 
 
If has rank 2, i.e. its null space is
of dimension 2, then two linearly independent
eigenvectors to 0.
, we need 2 linearly independent principal
vectors. In this case, the Jordan bas
A I n
A I q
Thus n
 

 

   
 
1 2 1 2
1
1 2
is takes the form
, , , and , , , , .
, is transformed into the Jordan canonical form
, .
k lv v v u u u k l n
So A
P AP diag J J
 

 
2026 Prof.Jiang@ECE NYU 335
Exercise 1
Find a transformation matrix to bring
the following matrix
1
, 0
0 1
into the Jordan Canonical Form
1 0
.
1 1
P
b
M b
J
    
    
Exercise 2
2026 Prof.Jiang@ECE NYU 336
1 1 1 1
3 3 5 4
8 4 3 4
15 10 11 11
A
           
Find a transformation matrix to bring the following
matrix into a Jordan form:
Solution:
2026 Prof.Jiang@ECE NYU 337
0 1 0 1
1 5 0 5
,
0 4 1 5
1 11 0 12
1 1 0 0
0 1 1 0
.
0 0 1 0
0 0 0 1
P
J
        
       
2026 Prof.Jiang@ECE NYU 338
 3
becomes (after elementary operations on rows and columns:
1 0 0 0
0 1 0 0
.
0 0 1 0
0 0 0 1
, the matrix has two elementary divisors:

I A
Therefore




       
 3
1 2
1 and 1 ,
which give two Jordan blocks, respectively:
1 1 0
1, 0 1 1 .
0 0 1
J J
  
        
See (the 1960 book of Gantmacher, Vol.1, pp.160-164)
for the details.
2026 Prof.Jiang@ECE NYU 339
Practicing Problems for Midterm
1. Compute the eigenvalues of the matrix
7 2

4 1
and transform it to one of the canonical forms.
A
    
2026 Prof.Jiang@ECE NYU 340
Practicing Problems for Midterm
1
1 2
2
1 2
2. Consider the block diagonal matrix
0
, with , .
0
Show that the eigenvalues of are those of
and .
i in n
i
A
A A n n n
A
A
A A
      

2026 Prof.Jiang@ECE NYU 341
Practicing Problems for Midterm
1 1
1
3. Assume is a nonsingular matrix. If
is an eigenvalue of with eigenvector ,
show that is an eigenvalue of .
In addition, give an eigenvector associated
with .
A
A x
A



 

2026 Prof.Jiang@ECE NYU 342
Practicing Problems for Midterm
0 1
4. Show that cannot be transformed into
0 0
a diagonal matrix under any similarity transformation.
A     
2026 Prof.Jiang@ECE NYU 343
Practicing Problems for Midterm
5. For any given 2 2 real orthogonal matrix
, one of the following must hold:
cos sin
(i) for some ;
sin cos
0 1 cos sin
(ii) for some .
1 0 sin cos
(Only for those
U
U
U
   
   

    
       
who love math proof!)
2026 Prof.Jiang@ECE NYU 344
Practicing Problems for Midterm
6. Show that is similar to . That is,
0 1 0 1
.
1 0 1 0
T
T
J J
J J
                
 
     
 
2026 Prof.Jiang@ECE NYU 345
Practicing Problems for Midterm
1 1 1 1
1
7. Assume that , are invertible matrices.
Show that
.
0 0
A D
A B A A BD
D D
   

         
2026 Prof.Jiang@ECE NYU 346
Practicing Problems for Midterm
 
1 1 1 1 1
1
11
8. Assume that , are invertible matrices.
Show that

where is the inverse of the
of
Schur compleme t
.
n
:
A D
A B A A BECA A BE
C D ECA E
E
A E D CA B
    


           
 
Note: A Very Useful Identity.
2026 Prof.Jiang@ECE NYU 347
Practicing Problems for Midterm
2 2
2 2
9. Reduce the following matrix into a
canonical diagonal form:
0

0

0 1

1 0
M
A
M
where
M


    
    
2026 Prof.Jiang@ECE NYU 348
Practicing Problems for Midterm
10. Reduce the following matrix into a Jordan
canonical form:
3 2 1
0 3 0
0 0 3
A
      
2026 Prof.Jiang@ECE NYU 349
Practicing Problems for Midterm
   
11. Rank Inequalities (See Horn-Johanson text, page 13)

, , we have
rank rank rank min rank , rank .

Sylvester inequality
Frobenius inequal
, , , we
ity
hav
m k k n
m k k p p n
A B
A B k AB A B
A B C
 
  

  
   

   
 
   e
rank rank rank + rank
with equality iff there are matrices and such that
.
AB BC B ABC
X Y
B BCX YAB
 
 
2026 Prof.Jiang@ECE NYU 350
Homework #7
 
1. For the matrix
1 0 1
0 2 0 ,
0 0 1
identify the spaces and the principal
vectors of grade 2.
g
A
P
      

2026 Prof.Jiang@ECE NYU 351
Homework #7
2. Express the following vectors as unique
representations of principal vectors found
in Problem 1:
2 0
9 , 9.3 .
84 0
x x
                
2026 Prof.Jiang@ECE NYU 352
Homework #7
3. Can you transform the following matrix into
a Jordan form:
0 , 0?
0 0
A
           

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