程序代写代做代考 C Lecture VII

Lecture VII
• The Jordan Canonical Form • Examples and Applications
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Review of Canonical Forms
 If A is an n  n matrix with distinct eigenvalues, then there exists a nonsingular matrix P s.t.
P1 AP  diag i .
 If A is Hermitian (possibly having repeated eigenvalues),
 a unitary matrix U such that U*AU diag .
i
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A Motivating Example
As stated previously, not every matrix can be transformed into a canonical diagonal form. For example,
A1 b, b0 0 1

cannot be transformed into a diagonal matrix. Fall 2020 Prof.Jiang@ECE NYU 297

Jordan Canonical Form
Theorem (Jordan):
Let A be an n  n matrix whose different eigenvalues
are  , ,  with multiplicities m ,, m : 1s 1s
    i.e.,  nonsingular P such that
s
 i1
m
det I  A 
Then, A is transformable into a Jordan canonical form.
P1AP  blockdiagi  J
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i i

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i
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Theorem (Jordan), cont’d: P1AP  blockdiagi  J
where
i 0  0
1   0 i
i        00 1 0
Jordan Block
i 0 0 1 

Comments
 In some texts, J T is used as Jordan form.  Different Jordan blocks, say i ,  j may be
associated with the same eigenvalues.
s ThetotalnumberofJordanblocks:ss n.
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Illustration via 3×3 matrices If a 33 matrix A has an eigenvalue 1
of multiplicity three, then it may be reduced into one of the following Jordan forms:
1 0 0 1 0 0 J0  0,J0  0,
1121 00 01
11 1 0 0
J1  0. 31
01 1

Remark 1
The distinct Jordan forms Ji , Jk , i  k, are not similar to each other.
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Remark 2
When each Jordan block  ( ) in the Jordan ii
form J is one-dimensional (i.e. n  1) and s  n, i
the Jordan matrix J becomes diagonal.
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Application to Matrix Analysis of Differential Equations
Given a set of 1st-order differential equations

x(t)Ax(t), x(0)n,
applying the transformation y  P1x yields: 
y(t)P1APy(t) : Jy(t).
y1 
 yi(t)yi(t),yimi,y .
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i
ys  

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 P resulting in Jordan form, let’s introduce the notion of principal vector, or generalized eigenvector,
which is a generalization of eigenvector.
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Principal Vectors
A (possibly zero) vector p is a principal vector of grade g  0 belonging to the eigenvalue i if
iIAg p0,
for which g is the smallest non-negative integer.
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Examples
• The vector p = 0 is the principal vector of grade 0.
• The (nonzero) eigenvectors are the principal vectors of grade 1.
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Motivating Question
In case of transformation to diagonal canonical form, i.e., P1 AP  diag i ,
the columns of P are linearly independent eigenvectors.
What about the matrix P in Jordan form?
How to construct P from principal vectors? Fall 2020 Prof.Jiang@ECE NYU 309

gi
i
Linear Spaces
Define the linear space composed of all principal vectors of grade  g belonging to i :
Pp|IAp0 g 
i.e., the null space of i I  Ag . Clearly,
P   P   P   0i1i2i
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An Interesting Result
Let A be an n  n matrix with the distinct
eigenvalues 1, , s , 1 s  n, with
multiplicities m , , m . 1s
Then, every vector x  n can be written as x  p1  p2  ps
where pi is a uniquely defined principal vector associated with i of grade  mi .
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Comment 1
A special, but interesting, case is when there are n linearly independent eigenvectors, say,
c1, , cn. In this case, i scalars s.t. xc1cn :p1pn.
1n
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Comment 2
Its proof relies upon the well-known Cayley- Hamilton theorem; see any standard matrix or linear algebra textbook.
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1 0 0 Consider the matrix A  7 1 0.
Example

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 can be written
as a unique representation of principal vectors.
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Answer
  1 m  2,   2 m 1. 1122
 The eigenvectors of 1 are of the form col0,1,0,  any nonzero scalar.
The eigenvectors of 2 are of the form   col 0, 0,1,  any nonzero scalar.
 P p2|p2col0,0,.
P p1|p1col,,0
21 12


Cayley-Hamilton Theorem Revisited
For any n  n matrix A,
(A)An 1An1n1AnIO
where  is the characteristic polynomial of A, i.e.,
detIAn n ini. i1
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 () 2 23. A
Example
ConsiderA7 4.Verifythat 8 5

1) The characteristic polynomial A () is:
2)(A)A22A3I00 0.
A
0 0 

Another Proof
Define the n  n matrix of signed cofactors: C   cof I  A.
T
Then, usingMcofM (detM)I,
IACT I. In addition,
CT n1C0 Cn1 Cn
for constant matrices Ci ‘ s.
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C0 I
C  AC   I
101

Proof (cont’d)
By identification of the coefficients of equal powers of  gives
Cn1  ACn2  n1I ACn1 nI.
Multiplying the first eq. by An , the second by An1,…,
and then adding them up leads to: O  AI.
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Question
How to compute principal vectors for a given matrix?
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A Motivating Example
Consider a 22 Jordan block J  0. 1 
Denote P  x1 x2  that transforms A into J. 
Namely, P1AP  J. So, we have AP  PJ, or
Ax1 x2x1 x2 0    1  
 Ax2  x2 , so x2 is an eigenvector;
(AI)x1  x2, so x1 is a principal vector (of grade 2). Fall 2020 Prof.Jiang@ECE NYU 321


Comment
Usually, x , x is called a Jordan Basis for 1 2
this 22 matrix A. In order words,
the JCF transformation matrix P is composed
of a Jordan basis, or a set of linearly independent eigenvectors and principal vectors.
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General Procedure
Step 1: Solve the characteristic equation AIz1 0.
Step 2 : For each independent z1 , solve AIz2 z1
2 wherez2 clearlysolves AI z2 0.
Collect only those z2 which are linearly independent
with the previously found eigenvectors z1.
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General Procedure
Step 3: For each independent z 2 , solve AIz3 z2
3 wherez3 clearlysolves AI z3 0.
Collect only those z3 which are linearly independent with the previously found vectors z1, z2.
Step 4: Continue in this way till the total number of independent eigenvectors and principal vectors equals
to the (algebraic) multiplicity of .
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General Procedure
Step 4 (cont’d): Denote
1 2 mm m1 1
x,x,,xz,z ,,z and
P  x1, x2 ,, xm . 
eigenvector
Therefore,
P1 AP  J (associated with eigenvalue ).
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Comments
• Not any arbitrary choice of linearly independent principal vectors would lead to a correct transformation matrix P.
For example, at Step 2, the linearly independent principal vectors z2 are chosen according to
AIz2 z1 but NOT :
IAz2 z1.
•
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See (the 1960 book of Gantmacher, Vol.1, Chap. VI, Section 8) for another general method of constructing a transformation matrix.

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x4 :AIx3 
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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 v of grade g  n  4. Define:
x1 : v
x2 :AIx1
 x3 :AIx2
Jordan Basis


That is,
eigenvector
Jordan Basis (cont’d)
Then, the 44 matrix A can be transformed into the Jordan canonical form:
 0 0 0
100 J  010
Principal vector of grade 4
0 0 1  
P1APJ, Px1,x2,x3,x4.
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Comment
If we define P  x4,x3,x2,x1 , then A is transformed 
into the Jordan canonical form J T , i.e.:  1 0 0
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010 1T 
P APJ 0 0  1. 0 0 0 


A More Complex Case
If AI has rank n2, i.e. its null space is of dimension 2, then  two linearly independent eigenvectorsto AIq0.
Thus, we need n  2 linearly independent principal vectors. In this case, the Jordan basis takes the form
v1,v2,,vk and u1,u2,,ul , kln. 
So, A is transformed into the Jordan canonical form P APdiag J1,J2 .
1 
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Exercise 1
Find a transformation matrix P to bring the following matrix
M1 b, b0 0 1

into the Jordan Canonical Form J1 0.
1 1 

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Exercise 2
Find a transformation matrix to bring the following matrix into a Jordan form:
1 11 1
3 3 5 4  A  8 4 3 4 
15 10 11 11 

Solution:
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0 1 0 1 1 5 0 5
P , 0 4 1 5
1110 12 
1 1 0 0
0 1 1 0 J  .
0 0 1 0 0 0 0 1


I  A becomes (after elementary operations on rows and columns: 100 0
0 1 0 0   .
0 0 1 0  
 3 0 0 0 1 
Therefore, the matrix has two elementary divisors:
 1 and  13 ,
which give two Jordan blocks, respectively:
1 1 0 J1,J0 1 1.
12 0 0 1

See (the 1960 book of Gantmacher, Vol.1, pp.160-164)
for the details.
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Practicing Problems for Midterm
1. Compute the eigenvalues of the matrix A7 2
41 
and transform it to one of the canonical forms.
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Practicing Problems for Midterm
2. Consider the block diagonal matrix A0
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A 1 ,withAnn,nnn. iii12
0A 2
Show that the eigenvalues of A are those of
A and A . 12

Practicing Problems for Midterm
3. Assume A is a nonsingular matrix. If  is an eigenvalue of A with eigenvector x,
showthat1 isaneigenvalueofA1.
In addition, give an eigenvector associated
with 1.
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Practicing Problems for Midterm
4. Show that A   0 1  cannot be transformed into 0 0

a diagonal matrix under any similarity transformation.
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Practicing Problems for Midterm
5. For any given 2  2 real orthogonal matrix U , one of the following must hold:
(i) U   cos sin  for some ; sin cos

(ii)U0 1cos sinforsome.
1 0sin cos  
(Only for those who love math proof!)
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Practicing Problems for Midterm
6. Show that JT is similar to J. That is, 01 01
  JT  J. 
10 10 
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Practicing Problems for Midterm
7. Assume that A, D are invertible matrices. Show that
 A B 1  A1 A1BD1  0D0 D1 .

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Practicing Problems for Midterm
8. Assume that A, D are invertible matrices. Show that
A B1 A1 A1BECA1 A1BE C D  ECA1 E 
 
where E is the inverse of the Schur complement 1
ofA: EDCA1B .
Note: A Very Useful Identity.
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Practicing Problems for Midterm
9. Reduce the following matrix into a canonical diagonal form:
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A M 022  0 M
 22  where
M0 1 1 0


Practicing Problems for Midterm
10. Reduce the following matrix into a Jordan canonical form:
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3 2 1 A0 3 0

0 0 3 

Practicing Problems for Midterm
11. Rank Inequalities (See Horn-Johanson text, page 13)
 Sylvester inequality Amk, Bkn, wehave
rankArankBkrankABminrankA, rankB.  Frobenius inequality
Amk, Bkp, Cpn, wehave
rankAB  rankBC  rankB + rankABC
with equality iff there are matrices X and Y such that
B  BCX YAB.
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1. For the matrix
1 0 1
Homework #7
A0 2 0, 
0 0 1 
identify the spaces P and the principal g
vectors of grade 2.

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Homework #7
2. Express the following vectors as unique representations of principal vectors found in Problem 1:
 2 x9
0  , x9.3.


84 
0  

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Homework #7
3. Can you transform the following matrix into a Jordan form:
  
A0  , 0? 
0 0  