A method of reducing matrices to the Jordan normal form

A METROD OF REDUCING MATRICES TO TRE JORDAN NORMAL FORM* G. A.

BEKISHEV

Novosibirsk 11

(Received

THE method

proposed below elementary collineatory

ing

Jordan

form and the

and is

convenient

1. Matrices

is

based on the transformations.

transforming

for

use

obtained

December

matrix

1964)

idea of systematically The method enables

to be constructed

employboth the

effectively,

on computers.

from the

unit

matrix

E by one of

the

following

operations: (a)

multiplying

(b)

interchanging

(c)

replacing

scribed Tij,

the

i-th

row (column)

the- i-th

and j-th

au element

as elementary.

Ui

We shall

j

=

by a number m # 0, rows

~columns),

in E by the number RL. will

0

denote

the

be de-

elementary

matrices

by Cifni),

and their

inverses

are

and Hij (m) respectively. The elementary

elementary;

matrices

are non-singular

also

in fact,

c-5 1

Ci-‘(m)=C(

Tij-’

(

=

T
E*j-l(m)=

Hij(-_m).

m

The elementary matrix

collineatory

A means that

j-th

column,

row.

The two other

Tij-‘ATij

‘t

Zh.

are

vychisl.

its

and its

j-th

transformation column,

row,

elementary

similarly

Mat.

i-th

Hij-l(n)Aflij(m)

multiplied

by-m,

multiplied

by m, subtracted

collineatory

transformations

interpreted.

mat.

Fiz.

is

5,

4,

189

722

- 726,

19fifi.

on the added to from

its

its i-th

C;-l(m)ACi(s),

5.A.

190

The matrix

A and its

Jordan

is a non-singular matrix. as a product of elementary multiplication

elementary to

This

these

is

matrices

for

rally

found

an arbitrary

of

A to

(2)

reduction

of

the

Let

us consider A direct

of

to the

well-known

A be an n-th

to J by

amounts

when reducing

In view

to

its

of

Jordan

of

corresponding

A

to how

this,

tri-

the

proce-

form splits

natu-

columns

order

n -

we carry

Suppose

We form the

the

normal

Jordan

form.

transformations

for

to

the

the

and hl

Al and taking unit

matrix

general

on a knowledge

algorithm

is

re-

case. of

the

basically

cell

0 -As

an eigenvalue. in its

as the

we reduce

it

AI is

greater

than

, ...,

these

for

Al,

matrices

to

the

any

columns

the

quasitriangu-

1 and it

and so on.

successive

I-A-IA,,__LI'~=

eigenvalues).

We concolumn

remaining

rl,

of

first

En.

ssme procedure

quasitriangular

in the

based This

by substituting

hzEi Cz II

II

here, matrix.

matrix,

rl

the

difficult

following.

as a result

l-z-'Ail-2 = Ai are

of

1 of

out

we obtain

I,

collineatory

by the matrix

A

matrix,

to the

form is

square

matrix

belonging

On transforming form

triangular matrix

of

to the

order

a non-singular

-

A

can be indicated

and amounts

eigenvector

(k
matrix

elementary

algorithm

the

forn.

triangular

and eigenvectors

If

problem

two problems.

eigenvalues

angular,

where

C-lAC,

A can be reduced entire

no difficulties

triangular

But another

lar

that

Jordan

a similar

these

choice

a matrix

Let

=

two stages:

reduction

struct

by J

transformations.

normal

(1)

2.

connected

Hence the

to present

to the

reducing

into

ducing

we conclude only.

elementary

choice

angular dure

by them,

transformations

choose

form J are

In view of tbe fact that C can be renresented matrices, and in view of tbe special features

C

of

Rekishev

steps

is

not

tri-

Slethod

of

reducing

(i=1,2,

Ui = Ei_f + l?i

It is easily seen that the matrix the triangular matrix1 = r-‘AT.

191

matrices

.,.,

k; U~==ri).

r = UlL’2 . . . uk transforms

A into

Note 1. We e&n obviously substitute linearly independent eigenvectors belonging to the eigenvalues in the first columns of ri. This shortens the number of computational steps but deprives us of the elementary method of computing ri”. Note 2. Suppose that hl. hi, ..+* h, are distinct eigenvalues of A of multiplicities k I, . . . , k, respectively. It is easily seen that the computational process can be organized so that the triangular matrix I is

Z=

(Zik)

I*;

Zih = 0,

i >

with triangular cells Iii on the main diagonal, ki and its diagonal elements hi.

k, the

order

of

Iii

being

The cells Iik, k > i, in the matrix I = fl,,,) ln can be made zero by elementary collineatory transformations which leave iii unchanged. This process has to be carried out in rows working upwards and in columns from left to right by the same method: the element &,q (q > p) in I, in a row with hk and in a column with hi (k # i), is made zero by the operation

As a result

the matrix A will

zii i

be reduced to the quasidiagonal

matrix

I22 4- . . . i 1.8.

matrices. corresponding.to 3. Let QI, Ql, . . . , Q, = I be triangular the successive principal minors of I = C&q) ln (gpq = 0, p > q); J1 = Q1. J21 **a, J, = 3 are Jordan forms of the Qi. The principle of reducing the triangular matrix I to the normal Jordan form .l by means of elementary collineatory transformations consists in constructing the sequence of triangular matrices

similar

to the matrix

I.

192

Bekishev

i;.A.

The sequence (1) is constructed inductively. If the matrix Ii is constructed as follows. already been constructed,

Ii-1

has

Let Ji_f = 3,,(a,) where J, .(aj)

is

main diakonal

(each

the

Jordan uI is

box of

one of

,

Obviously,

i-1).

1

_?=1

order

the

1 Pi-1

4-. . . i J,, (ad,

nj = i -

“j

with

diagonal

the

number CXj on the

elements

1. The elements

of

311, $22, *.., the

i-th

column

of Ii-1 lying above the main diagonal will be denoted by 61. 62, as etSi,le Among these elements we shall distinguish those elements located in one row with 1 of the corresponding Jordan box Jnj(oj), and the elements (there will be 2 of then) which do not have this property. We denote the non-zero elelnents of the 1st kind by 6,,, . . . , F,,, and of the 2nd by (0 < r $1).

ci gq2. . . . * Q. 91’ appear respectively shall

also

in the Jordan boxes Jnj ,(oj,!,

assume that

Jordan box J

assuming here the rows ql,

the

of

,jk(CXjk)

92, . . . , q,

, . . , Jnj .Caj f).

element 6 qk( 1 < k 5 r) corresponds

greatest

order

njk.

Be

to the

The element Gqk will

play an

important role in future. Finally, we shall agree to use in future, in“collineatory transformation with matrix C”, the stead of the expression shorter expression “transformation C”. le

consider

two cases.

Case A The algorithm for transforming a1 = a2 = . . . = al = $ii. matrix Ii-1 to Ii for this &ase is as follows.

First of all. we make the elements forming the tr~sfo~ations

Further,

$1,

with the aid of the transformation

element 6 qK. In place now appear.

of the elements

In the next step we carry

(s

=

1,

2, . . . . k-

SP2, . . . , spat, zero by per-

Ci(l/Sqk)

6, , s # k.

we normalize

the elements

outs the transformations

H

H

Q---l 1, k + 1, . . . . r;

from the

0

1 = TSAR---1)

the

6q /6 s qk

Uethad of reducing

and make the elements

6,S/6,K

zero.

matrices

After

this.

mains for us to permute the corresponding more exact, to perform the transformations Ti-i,

As a result creased

of this

i, Tt-2,

i-1,

permutation

by 1, The form of

Ii

* *

the order

is fully

We shall not mention a point of a call the set of elements of the j-th located in the columns i t 1, i + 2, mentioned transformations now reduce (al) (nj

the i-tb

+ l)-th

the subrows 9k, 9& - 1, . .

. ,

it O&Y

re-

or to be

qL+2.

defined

by this

is in-

statement.

purely practical kind. We agree to row of the transformed matrix, . . . , n. the j-th subrow. The aboveto the following operations:

multiplied

subrow is multiplied

Ii

of the box Jnb(ojKI

by 6 ,, is added to the n1

subrow (i = 1, 2, . . . , ml;

(a21 the i-th (ag)

subrcw in Ii-l,

to obtain

rows and columns,

Tqk+t,

*I

193

by 6&9; q&

-

nj,

+

1,

are added respectively to the subrows qs, ‘9,“9&’ n. + 1 (s = 1, 2. . . . . k - 1. k + 1, . . . . r-1;

multiplied qs -

1,

by 9S -

.. ..

1s

(a4)

the elements

Spj(j

= 1, 2. . . .,

(as)

the elements

Sqj(j

# k)

placed

i-th

n) are made zero;

are made zero,

end the element

6,,

re-

by unity;

fag) the columns numbered q& + 1, . . . , i - 1, are permuted with the column, and a symmetrical operation performed for the rows.

We have so far explicitly ments Sq1 *** 6 9r * If this

assumed that the 6i include is not the case, it is clear

i.e. to obtain Ii it Is sufficient to perform in al. The matrix J1 will here have the form Ji = I,,(Ul) -i- . ..-4-LI(Ul)

non-zero that

the operations

indicated

iri{&i).

Case 5 At

least

one of the oj

(j

= 1. 2, . . . ,

1) is

not

eUual

ele-

to Pii*

G.A. Bekishev

194

Suppose, for example, the transformations

al f pii.

It is easily

seen that,

as a result

of

where (j = 1, 2, . . . , n, -

the elements

61, 62, . . . . 6,1 disappear.

If there are other oj # pii, make zero those of the elements the saae rows with corresponding

by using similar transformations we C~UI 61, 62, . . . , 6i-l which are located in boxes Jnj(ojl. As a result of these

transformations we either obtain the matrix Ii, or (after several boxes Jnj(oj)) we find ourselves in the conditions Let us illustrate the proposed parison from 11, p. 1341. Eraapl

c.

i),

To reduce

permuting of case A.

method by an example borrowed

for

com-

the matrix 1 -1

A=

1 --1 3 -5 4 8 --A 3 -4 15 -10 II --il 1

-3

to the normal Jordan form J. We construct

the tr~sforming

matrix

G.

In order to reduce the absolute value of the elements, we carry out the transformations HAa( ffAx(-11, HAi( After this we permute the columns and rows numbered 1 and 2, i.e. we perform the transformation 7’12. As a result we get the matrix LIl-lAUl (see below), where Ul = *Y,,(l)H,,(-l)H,l(l)rl2. The simplest way to reduce the matrix obtained to the triangular form is to transform it by the matrices T23 and Hhs(l). On this occasion. however, we shall not make use of our above remarks, but follow the algorithm described in Section 2. We employ A.N. Krylov’s method for finding an eigenvalue h and the corresponding vector x of the cell Al. Taking < 0, 0. 1 > as the initial vector, we easily find h = -1. x = < 1. 4. I >. In accordance with Section 2 we construct the transforming matrix U2 (see below), invert it, then subtract the matrix 1 = L’2-1Ul-1AU1U2 which

,Yethod

of

reducing

matrices

195

gives

I ’

1

--1

U;lAUl

=

-1 -_--

collineatory

on the

Section

3,

0101

--1 1-i 4 0 4 0 -1 O-l 0’ 0 0 0-i 0

z=

tions

oioo

01° 0 4 --1 -4 0 -2 0 I1

I

The further

1000

4

transformations

triangular

matrix.

are

Following

connected

with

the principle

in

we have --1

II = I,

I2

=

Is = I-I;;(1) I2 Ha (1) =

II,

i

0

0 0

Zd = J = TzC;’

The transforming

Since

all

culty

in evaluating

(e-1) Hit (-4)

matrix

the matrices

Acknowledgements.

for

valuable

ZsHa (-4)

Cd (A-1) TM =

4

0

-1

0

O-l 0 0

0

--1

0; --i oi ii

0 1

-4

Borevich

the opera-

described

-1

0

-1

--_..-__ 0 0

i

0 0

0

.

o[ --1

is

except

L’2 are

elementary,

there

is

no great

diffi-

C.

The author advice.

wishes

to

thank D. K. Faddeev

Translated

and 2. I.

by D.E.

Brown

REFERENCES 1.

GANTMAKHER.F.R. dat,

1953.

Matrix

theory

(Teoriya

matrits),

Moscow,

Gostekhiz-