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 2026 Prof.Jiang@ECE NYU 307 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.) 2026 Prof.Jiang@ECE NYU 310 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 2026 Prof.Jiang@ECE NYU 311 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             51作业君版权所有