THE ALGEBRAIC THEORY OF MATRIX
POLYNOMIALS
J. Dennis
Cornell University
J. F
Traub
Carnegie-Mellon University
#
R. P. Weber
Bell Laboratories
October, 1973
•i
J. Dennis was supported
GJ27528.
in part by National Science Foundation Grant
J. F. Traub was supported in part by the National Science Foundation
under Grant GJ32111 and the Office of Naval Research under Contract
NQ014-67-A-0314-0010, NR 044-422.
ABSTRACT
m
A matrix S is a solvent of the matrix polynomial M ( X ) = A X + . . 4 A
#
m
if M(S) = 0 where A^, X, S are square matrices.
In this paper w e develop
the algebraic theory of matrix polynomials and solvents.
W e introduce
generalized division and interpolation, investigate the properties of block
Vandermonde matrices, and study the existence of solvents.
W e study the
relation between the matrix polynomial problem and the lambda matrix problem, which is to find a scalar \ such that A \
U
m
+ A..\
1
m
V...4A
is singular.
m
In a future paper w e extend Traub's algorithm for calculating zeros
of scalar polynomials to matrix polynomials and establish global convergence properties of this algorithm for a class of matrix polynomials.
1.
INTRODUCTION
Let Ag,A.|,... jA^, X be n by n complex matrices.
M(X) = A X
0
n
is a matrix polynomial.
m
m
W e say
1
+ A X " +...+A
1
m
1
A matrix S is a right solvent of M ( X ) if
M(S) = 0.
The terminology right solvent is explained below.
times refer to right solvents as solvents.
solvent of M(X) if M ( W ) is singular.
where M ( X ) is monic
For simplicity w e some-
W e say a matrix W is a weak
W e will deal primarily with the case
(A^ = I ) .
W e are interested in algorithms for the calculation of solvents.
Since
rather little is known about the mathematical properties of matrix polynomials and solvents, w e develop that theory here.
W e extend division and
interpolatory representation to matrix polynomials and study the properties
of block Vandermonde matrices.
In a future paper w e shall show that a gen-
!
eralization of T r a u b s algorithm
(Traub [66]) for calculating zeros of a
scalar polynomial provides a globally convergent algorithm for calculating
solvents for a class of matrix polynomials.
W e shall also report
on the use of solvents to solve systems of polynomial equations.
elsewhere
Most of
the results of this paper as well as additional material first appeared as
a Carnegie-Mellon University-Cornell University Technical Report
Traub, and Weber
(Dennis,
[71]).
If the A. are scalar matrices, A. = a.I, then (1.1) reduces to
l
I
l
M(X) « a X
0
m
+ a ^ "
1
+...+ a ^
(1.2)
2
This problem has been throughly studied
(Gantmacher
[60, Chapter 5]) and
we have such classical results as the Cayley-Hamilton Theorem and the
Lagrange-Sylvester interpolation
theorem.
If X is a scalar matrix, X = XI, then (1,1) reduces to
M(XD = M(\) = A X
Q
m
+
1
A^" "
1
+...+ A .
(1.3)
m
This is called a lambda-matrix and has also been thoroughly
(Lancaster [66]).
studied
Unfortunately, both (1.2) and (1.3) are sometimes called
matrix polynomials but we shall reserve this name for (1.1).
A great deal is
of course known about another special case of ( 1 . 1 ) , the case of scalar polynomials
(n = 1 ) .
A discussion of what is known about matrix polynomials may
be found in MacDuffee [ 4 6 ] .
of a matrix has been analyzed
The special case of calculating the square root
(Gantmacher
[60]).
A problem closely related to that of finding solvents of a matrix polynomial is finding a scalar X such that the lambda-matrix M(X)
is singular.
Such a scalar is called a latent root of M(\).and vectors b and r are right
and left latent vectors, respectively, if for a latent root p, M(p)b = 0^
T
a n c
*
T
r M(p) » 0_ .
See Lancaster [ 6 6 ] , Gantmacher
[ 6 0 ] , MacDuffee [ 4 6 ] , and Peters
and Wilkinson [70] for discussions of latent roots.
The following relation between latent roots and solvents is well-known
(Lancaster [66]). A corollary of the generalized Bezout Theorem states that
if S is a solvent of M ( X ) , then
M(X) = Q(X)UX-S),
where Q(X)
is a lambda-matrix of degree m - 1 .
S is called a right solvent.
(1.4)
It is because of (1.4) that
Here the lambda-matrix M(X)
has mil
latent
3
roots.
From (1.4) the n eigenvalues of a solvent of M(X) are all latent
roots of M ( \ ) .
M(\).
The n(m-1) latent roots of Q(X) are also latent roots of
Thus if one is interested in the solution of a lambda-matrix
problem,
then a solvent will provide n latent roots and can be used for matrix deflation, which yields a new problem Q ( \ ) .
W e summarize the results of this paper.
In Section 2 w e introduce a
generalization of division for matrix polynomials and derive some important
consequences.
This generalization includes scalar division and the gener-
alized Bezout Theorem as special cases.
sion of block companion matrices.
Section 3 contains a brief discus-
In Section 4 we give a sufficient condi-
tion for a matrix polynomial to have a solvent.
In Section 5 we
introduce
fundamental matrix polynomials and give a generalization of interpolation.
In the final section we study the block Vandermonde matrix which has some
surprising properties.
4
2.
DIVISION OF MATRIX
POLYNOMIALS
2
Let M(X) = X
+ A^X + A^.
It might seem natural to define division
of M(X) by the linear matrix polynomial X+C^ by
2
X
+ AX + A
}
= (X+cpCX+Cg) + C .
2
3
This cannot be done since the right side is not a matrix polynomial.
W e introduce generalized division for the class of matrix polynomials
such that the class is closed under this operation.
It reduces to scalar
division if n = 1.
m
Theorem 2.1
P
W(X) = X
m-1
Let M(X) = X
+ A X
1
+ ••• + A
and
m
P-l
+BX
1
+ ••• + B , with m > p.
p
Then there
exists a unique monic matrix polynomial P(X) of degree m-p
3
and a unique matrix polynomial L(X) of degree p-l such that
M(X) = F(X)X
Proof:
Let
P
P
+ B P(X)X "
1
+ ... + B P(X) + L(X). (2.1)
1
m
F(X) = X ~
L(X) = L X
o
p - 1
p
1
+ F^ ^"
P
+ L X "
X
2
1
+ ... +
+ ••• + L
and
m-p
Equating
5
coefficients of equation (2.1),
L L^
o>
P
P
1
#
3 2'* *
> F
a
m-p
n
d
• • • j L p ^ can be successively and uniquely
y
determined from the m equations.
QED
Equation (2.1) is the matrix polynomial division of
M(X) by W(X) with quotient P(X) and remainder L(X).
Definition 2.1
M(X) = X
m
Associated with the matrix polynomial
111
1
+ A-jX "" +
+ A ,
5
is the commuted matrix poly-
nomial
fl(X) = X
m
+ X^A,
+•••.+
.
A
(2 2
1
m
1
, C i
A
If
M(R) = 0,
then R Is a left solvent of M(X).
An important association between the remainder,
L(X), and the dividend, M ( X ) , in equation (2.1), will now be
given.
It generalizes the fact that for scalar polynomials
the dividend and remainder are equal when evaluated at the
roots of the divisor.
Corollary 2.1
If R is a left solvent of W(X), then
£(R) = M(R).
Proof:
Let
Q(.X) = M(X) - L(X). Then, it is easily shown
that
m
P
m
p
1
Q(X) 5 X " W ( X ) + X " " w C X ) P
1
+
+ ^(X)P ^ .
m
p
(2.3)
6
The result then follows immediately since Q(R) = 0 for
all left solvents of W ( X ) .
QED
The case where p = 1 in Theorem 2.1
this paper.
is of special importance in
Here w e have W ( X ) = X - R where R is both a left and right
solvent of W ( X ) .
Then Theorem 2.1
shows that
M(X) = F(X)X - RF(X) + L
where L is a constant matrix.
L = M(R) ,
(2.4)
Now Corollary 2.1 shows that
and, thus,
M(X) = F(X)X - RF(X) + M(R).
(2.5)
There is a corresponding theory for M(X) .
In this
case, equation (2.1) is replaced by
P
P
1
M(X) E X H(X) + X ~ H ( X ) B
1
+ ••• + H(X)B
p
+ N(X)
(2.6)
and Corollary 2.1 becomes the following.
Corollary 2.2
If S is a right solvent of W(X) , then
N(S) = M(S).
We again consider the case of
W(X) = X - S.
p = 1.
Let
Then equation (2.5) becomes
M(X) = XH(X) - H(X)S
+ M(S).
Restricting X to a scalar matrix XI, and noting that
(.2.7)
7
M(X)
= #(X),
we get the generalized Bezout Theorem
Chapter 4]) from equations
(2.5) and
(see Gantmacher [60,
(2.7):
M(X) = (IX-R)F(X) + M(R) = H(X)(IX-S) + M(S)
for any matrices R and S.
(2.8)
If in addition R and S are left
and right solvents, respectively, of M(X), then
M(X) = P(X)X - RP(X),
(2.9)
M(X) = XH(X) - H(X)S
(2.10)
M(X) = (IX-R)F(X) E H(X)(IX-S).
(2.11)
and
Hence, Corollaries 2.1 and 2.2 are generalizations of
Bezout
the generalized
Theorem.
Equation (2.11) is the reason why R and S are called left and
right solvents, respectively.
The use of block matrices is fundamental in this work.
It is use-
ful to have a notation for the transpose of a block matrix.
Definition 2.2
Let A be a matrix with block matrices
(B^^) of order n.
The
B (n)
block transpose of dimension n of A, denoted A
matrices
, is the matrix with block
(Bj^)•
The order of the block transpose will generally be
B (n )
dropped when it is clear.
except when
n = 1.
Note that, in general,
A
K
T
' ? A,
8
An important block matrix, which will be
studied in depth later in this paper, is the block
Vandermonde matrix.
Definition 2.3
given n b£ n matrices S , • • • , S ,
1
m
the block Vandermonde matrix is
v(s ,...,s ) =
1
m
A scalar polynomial exactly divides another
scalar polynomial, if all the roots of the division are al
roots of the dividend.
A generalization of the scalar
polynomial result is given next.
of Theorem 2.1,
The notation is that
9
Corollary 2.3
If W(X) has p left solvents, R, ,• • •,R
which
B
are also left solvents of M ( X ) and If V (R ,•••,R ) is non•i
P
singular then the remainder L(X) = 0.
3
1
3
Proof:
Corollary 2.1 shows that
i =
p.
£(R )
±
B
Since
= 0
#
V (R ,«* ,R )
1
for
is nonsingular,
and since
1
R,
RP"
R,
RP-l
P-l
J
£(R )
p-2
2
= 0
iP-1
R
L(R )
P
It follows that
MLX)
= FU)X
P
L(X) = 0.
P
+ B P(X)X "
1
1
Thus,
+ ••• + B F(X).
2
( -13)
QED
Prom equation (2.11) it follows that the eigenvalues of any solvent (left or right) of M(X) are latent
roots of M(X).
These equations allow us to think of right
(left) solvents of M(X) as right (left) factors of M(X).
In the scalar polynomial case, due to commutivity,
right and left factors are equivalent.
Relations between
left and right solvents can now be given.
10
Corollary 2.4
If S
and R
M(X), respectively, and S
then
F (S ) = 0,
where
are right and left solvents of
and R
have no common eigenvalues,
F(x) is
defined by
(2.9)
Proof:
Equation C2.9) shows that
F (S )S
Since S
and R
- R F (S ) = 0.
have no common eigenvalues,
F (S ) = 0
uniquely.
solution of
AX = XB
X = 0
3
(2..14)
This follows, since the
has the unique solution
if and only if A and B have no common
eigenvalues.
See Gantmacher
Given a left solvent R
that F (X) exists uniquely.
[60, C h a p t e r ^ .
QED
of M(X), Theorem 2.1 shows
If S is a right solvent of M(X)
and if F (S) is nonsingular (S is not a weak solvent of F (X)),
then equation (2,14) shows that
R
-1
= F (S)SF (S) .
( 2.15)
This gives an association between left and right solvents.
Corollary 2.5
and
If S and R are right and left solvents of M ( X ) , respectively,
if F(S) is nonsingular (S is not a weak solvent of F ( X ) ) , then
]
R = F(S) S F " ( S )
Proof:
This follows immediately from
(2.14).
(2.16)
11
3.
BLOCK COMPANION MATRIX
A useful tool in the study of scalar polynomials is the companion
matrix.
The eigenvalues of a companion matrix are the roots of its character
istic polynomial.
W e study properties of block companion matrices.
Defini-
tion 3.1, Theorem 3.1 and Corollary 3.1 can be found in Lancaster [ 6 6 ] .
Definition 3-1
Given a matrix polynomial
M(X) = X
m
1
+ A^X™'
+
+ A ,
m
the block companion matrix associated with it is
(3.1)
It is well known that the eigenvalues of the block
companion matrix are latent roots of the associated lambdamatrix.
Simple algebraic manipulation yields
rnn
m
m
1
Theorem 3-1 Det(C-XI) = (-l) det(lA +A A " +-•- + A ) .
1
m
Since C is an mn by mn matrix, we immediately obtain the following well known result.
Corollary
3.1 M(A) has exactly mn finite latent roots.
The form of the block companion matrix could have
been chosen differently.
Theorem 3,1 also holds for the
block transpose of the companion matrix:
12
(3.2)
I
-A
m
_A A
m
•A,
-1
It will be useful to know the eigenvectors of the
block companion matrix and its block transpose.
The results are a direct generalization of the
scalar case.
Theorem
See Jenkins and Traub [ 7 0 ] .
3.2 If p
±
is a latent root of M(X) and b^ and
right and left latent vectors, then p
and of C
B
i
are
is_ an eigenvalue of C
and
*1
(i)
B
is the right eigenvector of C ,
m-1,
P
i
^i
(ii)
is the left eigenvector of C, and
m-1
P
i
*i
13
_(m-l) |
b
(iii)
is the right eigenvector of C, where
(1)
(3-3)
Proof:
Parts (i) and (ii) are easily verified by substitutions into the appropriate eigenvalue problem.
For part (iii), consider
/ (m-l)l
d
-A
m
-A
—i
•
•
•
m-1
=
CL
( 1 )
p
i
—i
-A,
\
d
4
1
'
,(0)
(0)
\ -i
1
(3.4)
th
Multiply out; multiply the j
by
and add.
The result is
H (X)X - M ( X ) d j
0)
i
E
p ^ C A ) ,
where
m
1
m
2)
H .CA) 5 d j - > + d ^ " X + •
1
component equation
(3-5)
14
Substitution of \ =
M
p
^ i^-i°^
vector.
=
-
a n d
into Equation (3.5) shows that
>
h e n c e
l s
a
r
ight latent
Manipulating equation ( 3 . 5 ) , the result
equation ( 3 - 3 ) with
for
> ^±°^
j = l,*»«,m-l,
0
_d| ^ = b
follows.
±
and
d ^
=
QED
15
4.
STRUCTURE AND EXISTENCE OF SOLVENTS
W e establish a sufficient condition for a matrix polynomial to have
a solvent.
W e first establish some properties of a solvent.
From equation (2.11) it follows that the eigenvectors of a left
(right) solvent are left (right) latent vectors of the lambda-matrix.
Lancaster [66, p. 4 9 ] gives the characterization of a solvent that has only
elementary divisors.
Theorem
W e include the proof to help the reader.
4.1 If M(X) has n linearly independent right latent
vectors , b^, • • • *b * corresponding to latent roots P-^• • >P >
n
n
1
then Q A Q " is a right solvent, where
Q = [b^,. . . ,b ]
n
and
A = diag(p ,...,p ).
x
Proof:
From
n
1
M(QAQ" ) = ( Q A ^ A ^ A
result follows, since Q A
is just
M(p )b
±
1
= 0
1 1 1
" ^ * • •+A Q^Q^
M
M
111
1
the
1
+ A-.QA " + ••• + A Q
1
m
for i =!,•••,n.
QED
It follows from the above proof that if a solvent
1
is diagonalizable, then it must be the form Q A Q " , as in the
above theorem.
Corollary
4.1 If M(X) has mn distinct latent roots, and the
set of right latent vectors satisfy the Haar condition (that
every set of n of them are linearly independent), then there
are exactly (j^^
different right solvents.
17
Hence,
?
2
and P , the principal vectors of S,
satisfy equation
(4.1) , the definition of prin-
cipal latent vectors.
QED
The Fundamental Theorem of Algebra does not hold for
matrix polynomials.
This is known from the extensive
2
studies of the square-root problem:
Gantmacher feo,
p.231].
X =A.
See
Theorem 4.3 gives another matrix polynomial
which has no solvents.
Theorem 4*. 3
There exists a matrix polynomial with no sol-
vents .
Proof:
Consider
( 4.2 )
Det M(A)
four roots at
4X
3
+ 6X
X = 1.
solvent must either be
2
- 4X + i,
which has all
Thus, the Jordan form of a
J-. = I
or
J2
18
Since
M(I) ? 0,
it follows that J
feasible Jordan form.
2
M(l) =[
is the only
J
and, thus,
T
= (1,-1)
is the only latent vector, to within
a scalar multiple.
The second principal vector
f
is such that M (l)b + M(1)P = 0.
/2X-2
0 \
M (.X) =
and, hence,
\ 0
2X-2/
2
f
Thus,
P
2
= b
Here,
f
M ( D = 0.
to within a scalar multiple.
Using
Theorem 4.2 and the linear dependence of the first
two principal latent vectors, it follows that J
is
2
not a feasible Jordan form for a solvent of equation (4.2).
QED
Consider now the special case of a matrix polynomial whose associated
lambda-matrix has distinct roots.
We call a set of m solvents a complete
set if the mn eigenvalues of this set exactly match the distinct latent roots.
Note that w e have defined the concept of complete set only for the case of
distinct latent roots.
In Theorem 4.5 w e shall show that in this case a
complete set of solvents exists.
Lemma 4.1
We first prove two preliminary results.
If a matrix A is_ nonsingular, then there exists a
L
l
permutation of the columns of A to A such that
ll
A =[ ^
A
21
a
with A ^ and A
1
2 2
nonsingular.
A
A1 2 N
A
22,
18a
Proof:
Let A and
be matrices of orders n and k, re-
spectively, with arbitrary
lemma is false.
1 < k < n.
Assume the
Consider evaluating the deter-
minant as follows.
For each of the first k rows,
pick an element from a different column.
Then
19
multiply thes^e elements and the remaining minor.
The. sum, with appropriate signs, of every possible
choice of the k columns, is the determinant of A.
The k choices of the columns determine a square
matrix.
If that matrix is nonsingular, then the
minor must be zero, since the lemma was assumed
false.
Thus, such terms make no contribution to
the determinant of A.
A particular minor appears
several times in the sum.
It occurs the number
of ways the same k columns can be picked in different orders.
Each minor can thus be factored
from several terms; the result being the minor
times the determinant of the matrix formed by the
k columns and the first k rows.
Thus, if the
matrix formed by the k columns is singular, then
there is no contribution from this term in the
determinant of A.
Therefore, A must be singular,
which is a contradiction.
QED
Once the columns of A are permutated to get A ^
A
2 2
1
and
nonsingular, the process can be continued to similarly
divide
into nonsingular blocks without destroying the
nonsingularity of A ^ .
Theorem 4.4
If A, a matrix of order mn, is_ nonsingular, then
there exists a permutation of the columns of A to
with B^j a matrix of order n, such that
for
i = 1,•••,m.
B
A = ( j_j) >
is_ nonsingular
20
The existence theorem is given by
Theorem 4.5
If the latent roots of M(X) are distinct, then
M(X) has a complete set of solvents.
Proof:
If the latent roots of M(X) are distinct, then the
eigenvalues of the block companion matrix are distinct, and, hence, the eigenvectors of the block
companion matrix are linearly independent.
Theorem
3.2 the set of vectors
for
p
for
i = l,*«*,mn
Prom
m- 1,
i *i
are eigenvectors of C .
The
matrix whose columns are these mn vectors is nonsingular.
Theorem 4.4 shows that there are m dis-
joint sets of n linearly independent vectors b
-i
^
1
Using the structure QAQ
of Theorem 4 . 1 , the comx
plete set of solvents can be formed.
Corollary 4.2
QED
If M(X) has distinct latent roots, then it can
be factored into the product of linear lambda-matrices.
Proof:
Since M(X) has distinct latent roots, there exists
a right solvent S and
M(X) = Q(X)(IX-S).
Q(X) has
the remaining latent roots of M(X) as its latent
roots.
It follows then, that the latent roots of
Q(X) are distinct.
Thus, the process can be con-
tinued until the last quotient is linear.
QED
21
The process described in the above proof considers
solvents of the sequence of lambda-matrices formed by the
division
M(A) = Q(A)(IA-S).
C
Definition 4.2
#
A sequence of matrices ] _ > * ' >
c
f
o
r
m
chain
m
of solvents of M(X) if C. is a right solvent of Q ^ X ) , where
Q (X) = M(.X)
m
and
Q^(A) E Q^_^(X)(IX-C^)^
i=m,...,l.
It should be noted that, in general, only
right solvent of M(X). Furthermore, C
M(X) .
1
(4,3)
is a
is a left solvent of
An equivalent definition of a chain of solvents could
be defined with C., a left solvent of T.(X), and
T (X) E C l X - C
±
Corollary
If
4.3
m - 1 + 1
the latent roots of
)T
1 - ; L
M(\)
(X)
f
i = m,...,l. (4.4)
are distinct then
M(X)
has a chain of solvents.
If C ,---,C
1
m
m
m
form a chain of solvents of M(X), then
1
M(X) E I A + A A - + ••• A
1
+
M
E (IX- )(IX-C )---(IX-C ).
Cl
2
m
(4.5)
22
This leads to a generalization of the classical result for
scalar polynomials which relates coefficients to elementary
symmetric functions.
(4.5)
By equating coefficients of equation
one gets the following theorem.
Theorem 4.6
M(X) = X
111
If C, ,•••,0 form a chain of solvents for
1*
m
—
—
+ A.X "" + ••• + A , then
l
m
—
7
111
1
A
A
i
-
2 "
-(c .c
1
( 0
C
l 2
+ 0
a +
C
....c )
n
+
l 3 -"
+ 0
C
m-l m>
(
^
g
,
23
5.
INTERPOLATION AND REPRESENTATION
Given scalars s , •••,s , the fundamental
m
1
m
p
x
polynomials m. (x) =
( )>
, where p(x) = ~ T T ( X - s . ) ,
i
vx-s^;p \S^)
1=1
7
v
1
are
basic in interpolation theory.
Their
usefulness comes from the fact that m.(s.) = 6A A .
We
generalize these relations for our matrix problem.
Definition 5.1
Given a set of matrices, S_, ,
> 2
3
S , the
m
5
5
fundamental matrix polynomials are a set of m-1 degree
matrix
M
l
(
S
j
)
polynomials, M - ^ X ) ,
M ( X ) , such that
m
=
Sufficient conditions, on the set of matrices
S ^ , » ^ , S , for a set of fundamental matrix polynomials to
m
exist uniquely will be given in Theorem 5-2.
First, however,
we need the following results.
Theorem 5.1
Given m pairs of matrices , (X^Y. )
then there exists a unique matrix polynomial
i = 1, • • • ,m,
24
P(X) = A , X
+ A,X m-2 + ••• + A ,
m _ 1
for
PlX^) = y
t
i = ].,••• m ,
if and only if VCX.^ * * » X ) is nonsingular.
3
Proof:
such, that
m
1
3
P C X ^ = Y.^
for
'l
x
l
i =
m
• • •
I
• • •
X
Y
is equivalent to
•m
•
•
•
•
•
m-1
m
* m
1' 2*
• • •
m
QED
Corollary
5.1 Given m pairs of matrices, ( X ^ Y ^ ,
i = l,««»,m,
they uniquely determine a monic matrix polynomial
P(X) = X
m
+
A^x"
i = 1, •••,m,
Proof:
1 - 1
+
•••
+
A ,
m
such that
P(X ) = Y
±
for
±
if and only if V(X ,•••,X ) is nonsingular.
1
Let
Y
1
= Y - X^
±
m
and apply Theorem 5.1 to
±
(X $£.
±
qed
Let M(X) have a complete set of solvents. S-,,*«*,S .
I ' m
such that Y(S^ •••,S )
y
5.1,
m
is nonsingular.
According to Theorem
there exists a unique matrix polynomial
M.,(X) E A^h™-
1
1
J-
+ ... + A(i)
( j
<5.1 >
m
such that
(5.2 )
Note that M.(X) has the s ame solvents as M ( X ) , except S
been deflated out.
polynomials.
The M (X)
±
±
has
are the fundamental matrix
25
Denote by V ( S
jSj^x^i+i*
••*
1 >
#
1
s
* > ) the block
m
S
s
Vandermonde at the m-1 solvents, ]_> * * * * >
w
i
t
h
S
m
Theorem
£ deleted.
5 . 2 If matrices S ^ * • • , S are such that V C S ^ * • • ,S )
m
m
is nonsingular, then there exist unique matrix polynomials
M, (X) = A ^ X ™ "
i
1
1
+ ••• + A ^ ,
m
for
y
i =
9
'
9
m,
such that
M (X),•••,M (X) are fundamental matrix polynomials .
1
If,
m
furthermore, V(S ,«»*,S
1
#
k
s
j^^+i* * * > )
l s
nonsingular, then
m
Ck)
A^
lj3 nonsingular.
Proof:
V(S ,•••,S ) nonsingular implies that there exists
1
m
a unique set of fundamental matrix polynomials,
M ( X ) , • • •,M (X).
X
V(
m
S l
,•••,S _
k
nonsingular and Corollary
1 > S k + 1
,•••,S )
m
5.1 imply that there
exists a unique monic matrix polynomial
111
1
N CX) E X " + 4
k
)
k
that
N CSj) = 0
m
2
X "
for
k
j / k.
Q (X) ~= N ( S ) M C X ) .
k
k
k
Since
3
v
k
(
s
1
3 *
k
#
#
3 S
k )
,
such
Consider
Q (Sj) = N (
k
j = 1 •••,m.
+ ... + N ^
m
)
S j
)
for
is nonsingular
and both Q (X) and N (X) are of degree m-1, it
k
fc
follows that
N
S
Q (X) = N ( X ) .
k
M
k
X
N (X) = k ^ k ^ k ^ ^
-
R
cients, we get
I =
E
q
u a t l n
S leading coeffi-
W i ^
A
k
Thus,
a n d
t h u s A
[ ^
k
is nonsingular.
The fundamental matrix polynomials, M (X),
1
QED
(X),
m
can be used in a generalized Lagrange interpolation
formula.
Paralleling the scalar case we get the
following representation theorems.
26
Theorem 5.3
If matrices S,,•••,S are such that V(S,,•••,S )
J.
m
i»
> m'
m
is nonsingular, and M ^ S ) , • • • ,M (X) are a set_ of fundamental
m
matrix polynomials. then» for an arbitrary
m -
G(X) = B , X ^ +
+ B ,
m*
J-
(5-3)
it follows that
m
G(X) = £
G(S )M (X).
i=l
±
Proof:
Let
m
Q(.X) = ^
G(S )M (X).
i=l
for
i = l,^*,m.
±
i
(5 .4)
1
Then
Q(S ) = G(S )
±
±
Since the block Vandermonde is
nonsingular, it follows that Q(X) is unique and,
hence,
G(X) = Q(X)•
QED
A lambda-matrix was defined as a matrix polynomial
whose variable was restricted to the scalar matrix XI.
Thus,
the previous theorem holds for lambda-matrices as well.
Corollary 5.2
Under the same assumptions as in Theorem 5.3,
for an arbitrary lambda-matrix
G(A)
it follows that
5
B^
1
+ ... + B ,
m
(5.5)
27
m
G(X) = £
GCS^M^A).
(5.6)
i=l
Fundamental matrix polynomials were defined such that
M^(Sj) = 6^jI.
A result similar to equation (2.9) can
be derived based on the fundamental matrix polynomials.
It was previously (Section 2.1) developed using matrix
polynomial division.
Theorem 5.4 If M(X) has a set of right solvents,
S , - - , S , such that V ( S , - - , S ) - and V ( S ,
1
m
1
are nonsingular and
m
1
• • vVl^i+l >
(X),•••,M (X) are the set of fundam
mental matrix polynomials then
3
M (X)X
±
- S.M.CX)
i
83
e A
^
W
)
(5.7)
,
1,. • • ,m
1
where A ^ ^ is the leading matrix coefficient of M ( X ) .
1
±
## #
S
^ m
}
28
Proof:
Let
Q ( X ) = M (X)X - S ^ U ) . Note that
±
±
Q. (S.) = 0
for all j .
M(.X) is the unique monic
S
# #
matrix polynomial with right solvents ] _ > * >
since V C S ^ • • • ,S ) is nonsingular.
m
matrix coefficient of Q ^ X ) is
s
m
The leading
which is non-
singular, since V ( S , • • • ^ i . ^ S ^ , • • • ,S ) is
1
nonsingular.
Thus,
m
M(X) = A ^ '
Q (X).
±
QED
A previous re'sult (equation (2.5)) stated that if
R
is a left solvent of M(X) , then there exists a unique,
±
monic polynomial F ^ X ) of degree m-1, such that
M(X) E F (X)X - R ^ C X ) .
( 5,8)
1
Comparing equations (5.7) and C5• 8 ) , we obtain the following
result.
Corollary 5.3
F (X) = [ A ^ ]
i
Under the conditions of Theorem
M (X)
±
and_
is a left solvent of M(X).
5.4
29
If M(X) has a set of right solvents,
S ,---,S , such that VCS ---,S ) and VCS^- • s S ^ j S ^ p - • • S )
1
m
for
r
i = l,»*«,m
m
J
m
are all nonsingular, then, by equation
(5.9 ) , there exists a set of left solvents of
R
#
R
M(X), i * * * > »
s
u
c
h
t
h
a
t
m
Corollary
5.4
R
i
l s
similar to S
±
for all i.
Under the conditions of Theorem 5.4, if R
j
defined as in equation ( 5.9 . ) , then
Proof:
The result follows from equation (2.11) and
Corollary 5.3.
QED
s
30
6.
BLOCK
VANDERMONDE
The block Vandermonde matrix is of
fundamental importance to the theory of Matrix
Polynomials.
This section considers
some of its properties.
It is well known that in the scalar case (n = 1 ) ,
det V C , - - , s ) = II
S l
m
s
s
6
( i- j>
( -D
and, thus, the Vandermonde is nonsingular if the set of s ^ s
are distinct.
X
1
and X
2
singular.
One might expect that if the eigenvalues of
are disjoint and distinct, then V ( X X ) is non1 3
2
That this is not the case is shown by the follow-
ing example.
The determinant of the block Vandermonde at two
points is
r
I
I
det V ( X , X ) - det [
1
) = det ( X ^ X ) .
2
X
Even if X-^ and X
2
still be singular.
X
l
2,
have no eigenvalues in common, X
The example
X, = (
\-2
1
X
2
= I
J
yields X
2
- X
1
( 6.2)
singular.
2
j and
1/
- X
1
may
31
It will be shown that the X^ and X^ in this example
cannot be the set of solvents of a monic quadratic matrix
polynomial.
First
w e prove
Lemma 6.1 Let matrix A have distinct eigenvalues and N be a
3
subspace of c
then
n
of dimension d.
Av e N.
Suppose further that if
v e N>
Under these conditions d of the eigenvectors
3
of A are in N.
Proof:
Let
Av
= ^v
±
±
for
±
i = 1, • • • , ] ! •
The set of v ^ s
is a basis for c P since A has distinct eigenvalues.
3
Let 2
6
N
C C
n
and order the {v } such that
3
±
S3
V = ^
C
"i—i >
°± ^ 0
^
=
S
1 > •• • 5 •
Let
s
P(t) =
j = 2 ••• s.
3
Tf
J=2
(t-\.).
Then P(A)v. = 0 for
J
J
Hence
3
3
s
P(A)v_ = £
c P(A)v
1=1
1
= c^IT^
J=2
1
= c^CA)^
(A-Xjl)^ v
x
32
Let
0.
Thus v = -ji P(A)v e N.
—I
—
i = l,*'*,s.
Similarly, v. € N for
—i
The lemma follows, since
v. e N
was arbitrary.
Theorem
Q
E D
6.1 If M(X) has distinct latent roots, then there
exists a complete set of right solvents of M(X), S ,•••,S^,
1
and for any such set of solvents, ( p " * 3
v
Proof:
s
S
I N
2JL nonsingular.
)
The existence was proved in Theorem 4.5
.
The hypothesis
that S ,...,S , are right solvents of M ( X ) = X™ + A X'.m-1 + . +
1
is equivalent to
<V-> i>
I
I
s.
S
m
A
,m-l
>
m
(6.3)
Assume det V(S ,•••,S ) = 0,
1
and let N be the null
space of V(S ,«••,S ) . That is, v e N
if and
1
only if
V C S ^ - • • ,S )v_ = 0.
m
m
Since ^ > " ^
1
m
in
equation (6.3) exist, joining any row of
(-S ,»••»-S ) onto V(S ,»••,S ) gives a larger
1
m
1
matrix but with the same rank as V(S, .•••.S ) .
1'
'm
Thus, for all
all
v eN
v e N, ( i»' *' >SJJ)¥ = 0.
s
Hence, for
A
D
33
1
m
2
2
••
"m
v = V(S ,--^S )diag(S ,--,S )v.
0 =
1
m
1
m
,m
ra
(6.4)
Letting
A = diag(S •••,S^)
13
that for all _v e N
3
3
equation (6.4) shows
Av e N.
Since A has distinct
eigenvalues. Lemma 4.1 applies, and there are as
many eigenvectors of A in N as the dimension of N.
The eigenvalues of diag(S ••• S ) are the eigen13
3
f
values of the S^ s, and the eigenvectors are of the
T
T
T
form (0_ v 0 )
3
of the S^'s.
This is because if
3
3
where v is an eigenvector of one
0
v
y
0
then
S y = Xy
i
since
and
S^w = Aw.
This cannot be
and Sj do not have any common eigenvalues.
Let an arbitrary eigenvector of diag(S ,•••,S ),
1
(o ,v ,0 ) , be in N. Then
T
T
T
m
34
I
I
S.
S
/
\
0
= 0,
v
,m-l
But then,
,m-l
m
Iv = 0
0
which is a contradiction.
det V(S --- ,S ) ? 0.
13
Thus
QED
m
The example considered before this theorem was a
case where matrices X
eigenvalues and
1
and X
2
had distinct and disjoint
det V ( X , X ) = 0.
1
2
Thus, by the theorem,
they could not be a set of right solvents for a
monic, quadratic matrix polynomial.
In contrast with the
theory of scalar polynomials, we have the following result.
Corollary 6.1
There exist sets containing m matrices which
are not a set of right solvents for any
monic matrix polynomial of
degree m.
W e now prove a generalization of equation
(6.1), that the
Vandermonde of scalars is the product of the differences of
Let M•(d)
^,
Q ( X ) be a monic
l'-- k
the scalars.
U;
S
S
matrix polynomial of degree d >_ k with right solvents
S
l
3 # # #
S
* k*
T
h
e
s u
P
e r s c r i
P
t
d
will be omitted if
d = k.
Note that this matrix polynomial need not necessarily exist,
nor be unique.
Theorem 6.2
then
If V(S ,«»- ,S ) is nonsingular for
1
k
k= 2,**«,r-l,
35
det V(S ,.--,S ) = det V(S ,..«,S ^ ) det M
1
r
1
r
1
g
s
CS ).
r
(6.5)
Proof:
The nonsingularity of YiS •••,S
and Corollary
±s
5.1 guarantee that M
(X) exists uniquely
c
"r-1
1
The determinant of V C S ^ ' - . S ) will be evaluated
by block Gaussian elimination using the fact that
for an arbitrary matrix E of the proper dimension
A
B
A+EC
det
B+ED
= det
(6.6)
D
D
I
S
Det V(S ,»»»,S ) = det
1
r
,r-l
S
l
s --s
2
S
1
= det
Q
b
r-1 r-1
" l
S
r" l
Q
S
2
S
V 1
S
r
S
3" 1
1
S
V 1
= det
S
b
l 2
'
1
(6.7)
36
d
where
U ) = ( x - S * ) - (s^-S^CSg-S^^J^CX-S^^) .
n o n s i n
(^2"^1^
u
S l
a r
>
since
det (S^S-j^) = det V C S ^ S g ) t 0-
It will be shown
that after k steps of the block Gaussian elimination, the general term for the i,.j block,
1
i,j > k,
1
is M ^ " " ^ (S.). Assume it is true after k-1 steps
Then, after k steps, the i,j element Is
(1-1)
(
5
,
(1-1)
,
N (k-l)
s
M
1
1
This is merely M ^ " ^
(
s
r
l (k-l)
M
(X) evaluated at
s ---s
1
(
s
}
X = S..
j
k
Using the fact that the determinant of a block triangular matrix is the product of the determinants
of the diagonal matrices
(see Householder E64]),
the result follows.
Corollary 6.2 I_f V(S ,...,S ^) i£ nonsingular and
1
QED
is
k
not a weak solvent of M
Q
V
(X) , then V(S, , • • • ,S, ) is
Q
# b
1
k-1
K
nonsingular.
It is useful to be able to construct matrix polynomials with a given set of right solvents.
Corollary 6•3
Given ———————
matrices Sj_, ••• ,Sin —such
—————
— — that V(Sj_,•••,S,
i\. )
n
Is nonsingular for
K = 2,*'*,m,
1
the iteration
N (X) = I
N ( X ) = N _ (X)X - N _ ( S ) S N ^ ( S ) N _ ( X )
±
±
1
1
1
1
1
1
1
i
1
Q
(6-8)
37
i3 defined and yields an m degree monic matrix polynomial
W (X), such that N fS.) = 0 for i = l,«««,m.
m
Proof : N-^X) = X - S * M (X) . Assume N CX3 = M
m
1
x
g
fc
Then, from equation (6.8),
Nfc+iCS^ = 0
i = l,---,k+l
N,
+ 1
k
1
and, hence,
.. ,
g
(X) .
g
for
(X) = M«
~
(X) .
l
k+1
The sequence of block Vandermonde being nonsingular
+
guarantees the nonsingularity of
Corollary
6.4
S
m
s
1
S
S
]/ j_)*
If V t S ^ ' " ^ ) is nonsingular for
then 3_3 • • • ,S
M
#
S
Q
E D
k=2,***,m,
are a complete set of right solvents for
(X).
m
Proof:
The result follows directly from Theorem
5.4, where
we obtained
i)
(IA-S )M (X) = A ^ M ( X ) .
1
i
( 6.9)
38
BIBLIOGRAPHY
Dennis, Traub and Weber
[71]
Dennis, J., J. Traub and R. Weber, On
the Matrix Polynomial, Lambda-Matrix
and Block Eigenvalue Problems.
Cornell
University, Carnegie-Mellon University,
Computer Science Department technical
report, 1971.
[60]
Gantmacher, The Theory of Matrices.
I,
II, Chelsea Publishing Co., New York,
1960.
Householder[64]
Householder, A., The Theory of Matrices
in Numerical Analysis, Blaisdell, New
York, 1964.
Gantmacher
Jenkins and Traub
[70]
Jenkins, M . and J. Traub, A Three-Stage
Variable-Shift Iteration for Polynomial
Zeros and its Relation to Generalized
Rayleigh Iteration, Numer. M a t h . 14
1970, pp. 252-263.
Lancaster
[66]
Lancaster, P., Lambda-matrices and
Vibrating Systems. Pergamon Press, New
York, 1966.
MacDuffee
[46]
MacDuffee, C., The Theory of Matrices.
Chelsea, New York, 1946.
Peters and Wilkinson
Traub
[66]
[70]
Peters, G. and J. Wilkinson, AX = \BX
and the Generalized Eigenproblem.
SIAM
J. Num. A n a l . 7, 1970, pp. 479-492.
Traub, J., A Class of Globally Convergent
Iteration Functions for the Solution of
Polynomial Equations. Mathematics of
Computation, 2 0 , 1966, pp. 113-138.