SEPARATION OF SPECTRA IN BLOCK MATRIX TRIANGLES, SIMULTANEOUS SOLUTIONS OF SYLVESTER EQUATIONS AND THE PARLETT’S METHOD

SEPARATION OF SPECTRA IN BLOCK MATRIX TRIANGLES,
SIMULTANEOUS SOLUTIONS OF SYLVESTER EQUATIONS AND
THE PARLETT’S METHOD
SANG-GU LEE AND QUOC-PHONG VU∗

Abstract. For given k-tuples of commuting matrices (A1 , ..., Ak ) and (B1 , ..., Bk)
of dimensions m × m and n × n, respectively, we prove that the system of Sylvester
equations Ai X − XBi = Ci (i = 1, ..., k) has a simultaneous solution X such that
I X
Ai Ci
double commutes with
(for each k-tuple (C1 , ..., Ck) of
O O
O Bi
matrices of dimension m × n, which satisfy a natural compatibility condition), if
and only if the joint spectra of (A1 , ..., Ak) and (B1 , ..., Bk) are disjoint. We use
this result to obtain characterization of commuting block triangular matrices such
that the diagonal blocks have mutually exclusive joint spectra. We also present a
method of finding simultaneous solution of Sylvester equations, which is based on
Parlett’s recurrence relations.

Primary: 15A06, 15A21, 15A18,15A24, 15A27, 65F99 Sylvester equation; idempotent matrix; double commutant; block triangular matrix; Parlett’s recurrence relations; joint spectrum
1. Introduction
The Sylvester equation
AX − XB = C

(1.1)

is a popular subject in linear algebra and has many applications in control theory
and engineering. It was first proved by Sylvester [10] that if σ(A) ∩ σ(B) = ∅, then
there exists a unique solution X of equation (1.1). For a systematic account of results

related to Sylvester equations and applications, the reader is referred to [3].
In [7], the authors have extended Sylvester’s theorem to simultaneous solutions of
systems of Sylvester equations
AiX − XBi = Ci , (i = 1, ..., k),

(1.2)

Corresponding author.
The work by Quoc-Phong Vu is partially supported by the Vietnam Institute for Advanced Study
in Mathematics. The work by Sang-Gu Lee was supported by 63 Research Fund, Sungkyunkwan
University, 2012.
∗

1


where (A1, ..., Ak) and (B1, ..., Bk ) are commuting k-tuples of square matrices of dimensions m×m and n×n, respectively, and (C1, ..., Ck ) is a k-tuple of m×n matrices.
The k-tuple (C1 , ..., Ck) is called compatible (with (A1 , ..., Ak) and (B1 , ..., Bk )), if it
satisfies the following condition
Ai Cj − Cj Bi = Aj Ci − Ci Bj , (1 ≤ i, j ≤ k).

(1.3)

Recall that a point λ = (λ1 , ..., λk ) in Ck is called a joint eigenvalue of the k-tuple
(A1, ..., Ak) if there is a vector x ∈ Cm , x = 0, such that Aix = λi x, 1 ≤ i ≤ k.
The set of all joint eigenvalues of (A1, ..., Ak) is denoted by σ(A1, ..., Ak) and is called
the joint spectrum of (A1, ..., Ak ). For convenience of the reader, we restate the main
results of [7] as the following theorem.
Theorem 1.1. Let (A1, ..., Ak) and (B1 , ..., Bk ) be commuting k-tuples of m × m and
n × n matrices, respectively. The following are equivalent:

(i) σ(A1, ..., Ak) ∩ σ(B1 , ..., Bk) = ∅;
(ii) The system of homogeneous Sylvester equations AiX − XBi = O (i = 1, ..., k)
has only the trivial solution X = O.
(iii) For every compatible k-tuple (C1 , ..., Ck) of m × n matrices, the system of
Sylvester equations (1.2) has a unique simultaneous solution X.
Note that the compatibility condition (1.3) is equivalent to the fact that the maA i Ci
trices Ti =
(1 ≤ i ≤ k) are commuting. In this paper, we prove that
O Bi
condition σ(A1, ..., Ak) ∩ σ(B1, ..., Bk ) = ∅ is necessary and sufficient for the equations Ai X − XBi = Ci (1 ≤ i ≤ k) to have a simultaneous solution X such that
I X
FX =
is in the double commutant of {T1, ..., Tk }, i.e. FX S = SFX whenO O
ever STi = Ti S (for all i = 1, ..., k). As a consequence, we show that if T1, ..., Tk are
commuting block upper triangular matrices of dimension N × N


(i)
(i)
A1 A12 · · · A1p (i)

(i) 
 O A(i)
· · · A2p 
2
Ti = 
 (i = 1, ..., k),
 ··· ··· ··· ··· 
(i)
O

O · · · Ap
(i)

where the diagonal blocks Aj (j = 1, ..., p) are square matrices of dimension rj × rj
(with pj=1 rj = N), and Fj are N × N idempotent matrices of the form
Fj =

Ikj Xj
O O

j

, (where kj =

rl ),
l=1

2


(1)

(k)

such that Fj commutes with Ti , (for all i, j, 1 ≤ i ≤ k; 1 ≤ j ≤ p), then σ(Aj , ..., Aj )
are pairwise disjoint if and only if Fj double commutes with Ti , for all i and j.
Parlett [9] (see also [4]) gave a method of evaluating matrix functions F = f(T )
for upper triangular matrices with distinct eigenvalues which allows for a quick calculation of the super diagonal elements of F from the diagonal elements. In Section
1.3, we apply Parlett’s method for solving Sylvester equations (1.1). Moreover, we
show how Parlett’s method can be used for evaluating functions of several commuting matrices, as well as for finding simultaneous solutions of Sylvester equations (1.2).

(i)

(i)

In the sequel, Ai = (asj ) and Bi = (bsj ) (1 ≤ i ≤ k) denote square matrices of
(i)
dimension m × m and n × n, respectively; Ci = (csj ) and X = (xsj ) (1 ≤ s ≤ m, 1 ≤
j ≤ n) are matrices of dimension m × n, all matrices are over the field of complex
numbers. The identity matrix of dimension m × m is denoted by Im , and the zero
matrix of dimension m × n is denoted by Om,n , or simply by O (we write On for On,n ).
We will denote by Ti the square matrix of dimension N × N (N = m + n) defined by
Ti =

A i Ci
On,m Bi

.

(1.4)

If X is an m × n matrix, then we denote by FX the N × N matrix which has the form
FX =

Im X
O On

.

(1.5)

For a family of square matrices {T1, ..., Tk}, we denote by {T1, ..., Tk} the commutant of {T1 , ..., Tk}, i.e. {T1, ..., Tk} = {S : STi = Ti S for all i = 1, ..., k},
and by {T1, ..., Tk} the bicommutant (or double commutant) of {T1, ..., Tk }, i.e.
{T1, ..., Tk} = {V : V S = SV for all S ∈ {T1, ..., Tk} }.
2. Sylvester equations and block triangular matrices
First we observe that the following simple statement holds.
Proposition 2.1. Suppose that (A1, ..., Ak) is a k-tuple of m×m matrices, (B1 , ..., Bk )
is a k-tuple of n × n matrices and (C1, ..., Ck ) is a k-tuple of m × n matrices. Then
a matrix X is a simultaneous solution of the system of equations (1.2) if and only if
FX ∈ {T1, ..., Tk } , where Ti and FX are defined by (1.4) and (1.5), respectively
The proof of Proposition 2.1 is straightforward. It should be noted that, in Proposition 2.1, as well as in Theorem 2.2 and Corollary 2.3 below, the k-tuples (A1, ..., Ak),
(B1 , ..., Bk) and (C1 , ..., Ck) are arbitrary (there is no commutativity requirements on
(A1, ..., Ak), (B1 , ..., Bk ) and there is no compatibility requirement on (C1 , ..., Ck)).
3


Theorem 2.2. Let (A1, ..., Ak) be a k-tuple of m × m matrices, (B1 , ..., Bk) be a
k-tuple of n × n matrices, and (C1 , ..., Ck) be a k-tuple of m × n matrices, and let
Ti (1 ≤ i ≤ k) and FX be defined by (1.4) and (1.5), respectively. Suppose that
the system of equations (1.2) has a simultaneous solution X. Then the following are
equivalent:
(i) FX ∈ {T1, ..., Tk} ;
(ii) The homogeneous systems of Sylvester equations AiY − Y Bi = O and ZAi −
Bi Z = O have only the trivial solutions Y = O, Z = O.
Proof. First, assume that (ii) holds. By Proposition 2.1, we have FX ∈ {T1 , ..., Tk} .
We prove that, indeed, FX ∈ {T1, ..., Tk} , that is, FX S = SFX whenever S ∈
{T1, ..., Tk} .
Suppose a matrix S ∈ {T1, ..., Tk} has the following block forms conforming to the
structure of Ti :
S=

S1 S2
S3 S4

.

From STi = Ti S, we have
A i S1 + C i S3 = S1 A i ,

(2.1)

AiS2 + Ci S4 = S1 Ci + S2 Bi ,

(2.2)

Bi S3 = S3Ai ,

(2.3)

Bi S4 = S3 Ci + S4 Bi ,

(2.4)

for all i = 1, ..., k. From (2.3) and the the assumption that the homogeneous equation
ZAi − Bi Z = O have the only trivial simultaneous solution Z = O, it follows that
S3 = O. Hence, the identities (2.1)-(2.4) become
A i S1 = S1 A i ,

(2.5)

AiS2 + Ci S4 = S1 Ci + S2 Bi ,

(2.6)

Bi S4 = S4Bi .

(2.7)

We have, by (2.5) and (2.7)
Ai(S1 X − XS4 ) − (S1 X − XS4 )Bi
= S1 (Ai X − XBi ) − (Ai X − XBi )S4
= S1 C i − C i S4 ,

(2.8)

and by (2.6)
AiS2 − S2 Bi = S1 Ci − Ci S4 .
4

(2.9)


From (2.8) and (2.9) and the assumption that the system Ai Y − Y Ai = O have only
the trivial simultaneous solution Y = O, it follows that
S1 X − XS4 = S2 ,
which implies SFX = FX S.
Conversely, assume now that FX ∈ {T1, ..., Tk} . Let Y and Z be the simultaneous
solutions of equations AiY − Y Bi = O and ZAi − Bi Z = O, respectivley. We show
that Y = O, Z = O.
Let
GY =

O Y
O O

Then one can verify directly that GY ∈ {T1 , ..., Tk} . Hence GY FX = FX GY , which
implies Y = O. Analogously, let
−XZ −XZX
Z
ZX

HXZ =

.

Then it is straightforward to verify that HXZ commutes with all Ti (i = 1, ...., k),
hence FX HXZ = HXZ FX , which implies Z = O.
In particular, the case Ci = O for all i = 1, ..., k and X = O in Theorem 2.2 yields
the following corollary.
Corollary 2.3. Let (T1, ..., Tk ) be a k-tuple of N × N matrices which have the same
block diagonal structure Ti = Ai ⊕ Bi , where (A1, ..., Ak ) and (B1, ..., Bk ) are k-tuples
Im O
of m × m and n × n matrices, respectively, and let F =
.
O On
Then the following are equivalent:
(i) F ∈ {T1, ..., Tk } ;
(ii) The homogeneous systems of Sylvester equations AiY − Y Bi = O and ZAi −
Bi Z = O have only the trivial solutions Y = O, Z = O.
Remark 2.4. If the system of Sylvester equations (1.2) has a simultaneous solution
X, then the matrices Ti, defined by (1.4), are simultaneous similar to the matrices

(0)

Ti

=

Ai O
O Bi

.

Namely, if
V =

Im X
O In
5

,


(0)

then V −1 TiV = Ti for all i = 1, ..., k. Using this, one can first prove Corollary 2.3
and then deduce Theorem 2.2 from it.
If, in addition, the k-tuples (A1, ..., Ak) and (B1 , ..., Bk) are commuting, then we
have the following result, which follows immediately from Theorem 1.1, Theorem 2.2,
and Corollary 2.3.
Corollary 2.5. Let (T1, ..., Tk ) be a k-tuple of commuting N × N matrices which have
the same block diagonal structure Ti = Ai ⊕ Bi , where (A1 , ..., Ak) and (B1, ..., Bk )

are commuting k-tuples of m × m and n × n matrices, respectively, and let F =
Im O
. Then the following are equivalent:
O On
(i) σ(A1, ..., Ak) ∩ σ(B1 , ..., Bk) = ∅;
(ii) F ∈ {T1, ..., Tk } ;
(iii) For every compatible k-tuple (C1 , ..., Ck) of m × n matrices, there exists a
simultaneous solution X of equations (1.2) such that FX ∈ {T1 , ..., Tk} .
From Corollary 2.5 we obtain the following result by mathematical induction.
Theorem 2.6. Suppose Ti (1 ≤ i ≤ k) are commuting block upper triangular matrices
of dimension N × N which have the following form


(i)
(i)
(i)
A1 A12 · · · A1p

(i) 
 O A(i)
· · · A2p 
2
Ti = 
,
 ··· ··· ··· ··· 
(i)
O
O · · · Ap
(i)

where the diagonal blocks Aj (1 ≤ j ≤ p) are square matrices of dimension rj × rj
(with pj=1 rj = N), and suppose Fj (1 ≤ j ≤ p) are N × N idempotent matrices of
the form
Fj =

Ikj Xj
O O

,

j

where kj = l=1 rl , such that Fj ∈ {T1 , ..., Tk} . Then the following are equivalent:
(i) The joint spectra of the corresponding diagonal blocks are pairwise disjoint, that
(1)
(k)
(1)
(k)
is σ(Aj , ..., Aj ) ∩ σ(Al , ..., Al ) = ∅ for all 1 ≤ j, l ≤ k, j = l;
(ii) Fj ∈ {T1, ..., Tk} for all j = 1, ..., p.
Theorem 2.6, in turn, implies the following result about block diagonal matrices:
Suppose Ej (1 ≤ j ≤ p) are idempotent matrices of dimension N × N, with rj =
rank (Ej ) > 0, such that Ei Ej = O if i = j and IN = E1 +E2 +···+Ep. Then {Ej }pj=1 is
called a (non-trivial) resolution of the identity. Let (T1, ..., Tk) be a commuting k-tuple
6


of N × N matrices such that Ti commutes with Ej for all i, j (1 ≤ i ≤ k, 1 ≤ j ≤ p).
Then the matrices Ti (1 ≤ i ≤ k) have joint (i.e., simultaneous) block diagonal forms
corresponding to the decomposition CN = ran(E1 ) ⊕ ran(E2) ⊕ · · · ⊕ ran(Ep) (where

ran(E) denotes the range of E). Thus, Ti and Ej have block diagonal representations
of the same block structure:
(i)

(i)

Ti = diag(A1 , ..., Ak ) (i = 1, ..., k), Ej = diag(O, ..., O, Irj , O, ..., O) (j = 1, ..., p).
Applying Theorem 2.6 for Fj = E1 + ... + Ej (1 ≤ j ≤ p) (i.e. the case when
= 0 for all l < j and all i, and X = O), we get the following result.

(i)
Alj

Corollary 2.7. Suppose the matrices Ti (1 ≤ i ≤ k) and Ej (1 ≤ j ≤ p) have
(i)
(i)
block diagonal representations of the same block structure Ti = diag(A1 , ..., Ak ),
Ej = diag(O, ..., O, Irj , O, ..., O) (the j-th diagonal block of Ej is Irj , the other diagonal
blocks of Ej are zero matrices). Then the following are equivalent:
(1)
(k)
(i) The joint spectra of the corresponding blocks of Ti are disjoint, that is σ(Aj , ..., Aj )∩
(1)
(k)
σ(Al , ..., Al ) = ∅ for all 1 ≤ j, l ≤ p, j = l;
(ii) Ej ∈ {T1, ..., Tk } (for all j = 1, 2, ..., p).
The case k = 1 of Corollary 2.7 is contained in ([6], Theorem 4.1).
3. An algorithm for finding simultaneous solutions of Sylvester
equations
The question of finding efficient methods for computing functions of square matrices

plays an important role in linear algebra and its applications.
If T is a square matrix, then there is Schur decomposition T = QSQ∗ where S is
upper triangular and Q is a unitary matrix. Since f(T ) = Qf(S)Q∗, computation of
f(T ) is reduced to computation of f(S), so that, in effect, in computing functions of
matrices we can always assume that the matrices are upper triangular. For various
methods of computing matrix functions, the reader is referred to [1], [2], [4], [5], [9],
[8] and references therein. Note that the Schur decomposition also is valid for several
commuting matrices, in the sense that commuting matrices can be simultaneously
brought to upper triangular form, that is, if T1, ..., Tk are commuting matrices, then
there exists a unitary matrix Q such that Ti = QSi Q∗ where Si are upper triangular
for all i = 1, ..., k. This implies f(T1 , ..., Tk) = Qf(S1, ..., Sk )Q∗, where f(z1, ..., zk ) is
a function of k variables such that f(T1 , ..., Tk) is well defined. Hence, in evaluating
functions of commuting matrices we may assume that the matrices are upper triangular.
Analogously, if (A1, ..., Ak) and (B1, ..., Bk ) are commuting k-tuples of square matrices of dimensions m × m and n × n, respectively, then there exist unitary matrices
7


Q and R such that Ai = QAi Q∗ and Bi = RBi R∗ , where Ai and Bi are upper triangular. A matrix X is a simultaneous solution of equations (1.2) if and only if the
matrix X = Q∗XV is a simultaneous solution of equations AiX − X Bi = Ci , with
Ci = Q∗Ci V . Thus, in finding simultaneous solutions of Sylvester equations (1.2) we
may again assume that the matrices Ai and Bi are upper triangular for all i = 1, ..., k.
Parlett [9] gives an efficient method of evaluating matrix functions F = f(T ) for
upper triangular matrices T with distinct eigenvalues, which allows for a quick calculation of all (super diagonal) elements of F from the diagonal elements. Below we
extend the Parlett’s method to the case of functions of several commuting matrices
and apply it to the problems of finding simultaneous solutions of Sylvester equations.
In the sequel, we denote elements of F and T by flj and tlj , respectively (1 ≤
l, j ≤ N). Note that since T is upper triangular, F also is upper triangular, and
the diagonal elements of F are readily computed; namely, fjj = f(tjj ). Thus, it
remains to compute the elements flj which are above the main diagonal (i.e. for
l < j). Parlett’s method is based on the observation that the commutativity relation

F T = T F is equivalent to the following relations:
j−1

(fls tsj − tls fsj ), for all l < j,

flj (tll − tjj ) = tlj (fll − fjj ) +

(3.1)

s=l+1

(the summation in (3.1) is void if j = l + 1). Since tll = tjj (∀l = j), (3.1) implies the
following recurrence relation
j−1

1
flj =
[tlj (fll − fjj ) +
(fls tsj − tls fsj )] (l < j).
tll − tjj
s=l+1

(3.2)

From the relation (3.2) it follows that any element flj (l < j) can be calculated as
soon as elements on the same row to the left of flj (i.e., the elements fls with s < j)
and elements on the same column below flj (i.e. the elements fsj with s > l) are
known. Therefore, elements flj can be calculated a super diagonal at a time, starting
with the main diagonal fjj = f(tjj ).
Parlett’s approach can be used for solving Sylvester equation AX − XB = C for

the case A and B are upper triangular matrices. Namely, let X be a solution of
equation AX − XB = C, and let
T =

A C
O B

, FX =

I X
O O

.

Then FX is upper triangular. Moreover, although FX is an N × N matrix, with
N = m + n, there are only mn unknown elements of FX , namely the elements
8


flj = xl,j−m with 1 ≤ l ≤ m, m + 1 ≤ j ≤ N. The diagonal elements fll = 1 for
1 ≤ l ≤ m and fll = 0 for m + 1 ≤ l ≤ N, as well as all other remaining elements, are
zero. It should be emphasized that our situation is different from that described by
Parlett’s algorithm, since not all eigenvalues of T are necessarily distinct, as required
in the original Parlett’s method for evaluating matrix functions. However, due to the
structure of the matrix FX , with many super diagonal elements known a priori, we
still can apply the Parlett’s recurrence relations to evaluate the remaining elements
of FX (the elements flj with 1 ≤ l ≤ m, m + 1 ≤ j ≤ N). From (3.1) and taking into
account the fact that fls = 0 for l < s ≤ m and fsj = 0 for s ≥ m + 1, we have
j−1

flj (tll − tjj ) = tlj +

(fls tsj − tlsfsj )
s=l+1
j−1

m

(fls tsj − tlsfsj ) +

= tlj +

(fls tsj − tls fsj )

(3.3)

s=m+1

s=l+1
j−1

m

= tlj −

fls tsj (1 ≤ l ≤ m, m + 1 ≤ j ≤ N).

tls fsj +
s=m+1

s=l+1

Since flj = xl,j−m , tlj = cl,j−m for l ≤ m, j > m, tls = als for 1 ≤ l, s ≤ m,
ts+m,j+m = bs,j for 1 ≤ s, j ≤ n, from (3.3) we obtain the following recurrence
formula for the elements xlj of the matrix solution X:
j−1

m

xlj (all − bjj ) = clj −

xls bsj , (1 ≤ l ≤ m, 1 ≤ j ≤ n).

als xsj +
s=l+1

(3.4)

s=1

Now assume that σ(A) ∩ σ(B) = ∅, i.e. all = bjj for all l, j (1 ≤ l ≤ m, 1 ≤ j ≤ n).
From (3.4) we get the following recurrence formula for xlj .
m

1
xlj =
(clj −
als xsj +
(all − bjj )
s=l+1

j−1

xls bsj ), (1 ≤ l ≤ m, 1 ≤ j ≤ n).

(3.5)

s=1

Relations (3.5) allow us to compute the elements xlj of the solution matrix X recursively, one sub diagonal, middle diagonal and super diagonal at a time, starting with
element xm1 :
xm1
xm−1,1; xm2
xm−2,1; xm−1,2 ; xm3
.....
Now let us consider an extension of Parlett’s method to functions of several commuting matrices and apply it to finding simultaneous solutions of Sylvester equations
9


(1.2). First, we note that the Parlett’s method can be slightly modified so that it
can be used for evaluating functions of several commuting N × N matrices T1, ..., Tk,
for the case when Ti (i = 1, ..., k) are upper triangular matrices such that the joint
spectrum σ(T1, ..., Tk) consists of N distinct elements. Let Ti have the following
upper triangular form


(i)
(i)
(i)
(i)

t1 t12 t13 · · · t1N

(i)
(i) 
 0 t(i)
t23 · · · t2N 
2
Ti = 
 , (i = 1, ..., k).
 ··· ··· ··· ··· ··· 
(i)
0
0 · · · 0 tN
(1)

(k)

Then σ(T1, ..., Tk) = {λj := (tj , ..., tj ) : j = 1, ..., N}, and the condition λl = λj
for l = j is equivalent to:
For every l = j (1 ≤ l, j ≤ N), there exists p := p(l, j), 1 ≤ p ≤ k,
(p)

such that tl

(p)

= tj .

(3.6)

Let F = (flj )N
l,j=1 be a function of T1 , ..., Tk, F = f(T1 , ..., Tk). Then F is upper
triangular (so that flj = 0 for l > j), and the commutativity relation F Ti = Ti F
implies the following recurrence relations for element flj with l < j:
j−1
(i)
flj (tll

−

(i)
tjj )

=

(i)
tlj (fll

(i)

(i)

(fls tsj − tls fsj ) (1 ≤ l < j ≤ N, 1 ≤ i ≤ k).(3.7)

− fjj ) +
s=l+1

From conditions (3.6)-(3.7) we have
For all l < j (1 ≤ l, j ≤ N), there exists p := p(l, j), 1 ≤ p ≤ k, such that
(p)

tll

=

(p)
tjj

and flj =

j−1

1
(p)
(tll

−

(p)
tjj )

(p)
tlj (fll

(p)

− fjj ) +

(p)

(fls tsj − tls fsj ) . (3.8)

s=l+1

Remark 3.1. Each value flj must satisfy k relations (3.7) (for i = 1, ..., k) but is
computed using one value p such that (3.6) holds. However, it is consistent in the
sense that the relations (3.7) for all other values i (1 ≤ i ≤ k) are automatically
satisfied.
Formula (3.8) allows us to compute elements flj once the elements fls with s < j
(i.e., elements on the same l-th row to the left of flj ) and the elements fsj with
s > l (i.e., elements on the same j-th column below flj ) are known. Since fjj =
(1) (2)
(k)
f(tj , tj , ..., tj ) and flj = 0 (for all l > j) are known, Formula (3.8) allows us to
compute all elements flj with l < j a super diagonal at a time, starting from the
10


super diagonal {f12, f23 , ..., fN −1,N }. Thus, we obtain the following algorithm for calculating F = f(T1 , ..., Tk):
(1)

(2)

(k)

1. For each j = 1, ..., N, calculate fjj = f(tj , tj , ..., tj ).
2. For each j = 1, ..., N − 1
for each l = 1, ..., N − 1
(p)
(p)
find an element p := p(j, l) with 1 ≤ p ≤ k and tj = tj+l and calculate
fj,j+l =

j+l−1

1
(p)

(p)

(p)

(tj − tj+l )

(p)

[tj,j+l(fj,j − fj+l,j+l ) +

(p)

(fjs ts,j+l − tjs fs,j+l )]
s=j+1

Remark 3.2. The above method of evaluating functions of several commuting matrices
has merit on its own, even in cases when it seems possible to reduce to functions of a
single matrix. For example, it may happen σ(T1, ..., Tk) consists of N distinct elements
(in Ck ), but the matrix T = T1 + ... + Tk does not have distinct eigenvalues. Hence we
can use the above algorithm for evaluating f(T ), but the original Parlett’s algorithm
is not applicable.
Analogously, we can exploit the Parlett’s method for finding a simultaneous solution
of equations (1.2), for upper triangular Ai and Bi . Let X denote a simultaneous
(i)

solution of equations (1.2), with elements (xlj ) (1 ≤ l ≤ m, 1 ≤ j ≤ n). Let Ti = (tjs )
and FX = (flj ) (1 ≤ l, j ≤ N), be defined by (1.4) and (1.5), respectively. Note that
Ti (for all i = 1, ..., k) and FX are upper triangular. Moreover, for the elements
(fl,j )N
l,j=1 of FX we have

if l = j, 1 ≤ l, j ≤ m
 1
xl,j−m if 1 ≤ l ≤ m, m + 1 ≤ j ≤ N
flj =
(3.9)

0
for all other l, j

By Proposition 2.1, the matrix X is a simultaneous solution of equations (1.2) if
and only if FX Ti = TiFX (for all i = 1, ..., k). As above, the commutativity relation
FX Ti = TiFX implies the relations (3.7). From (3.7) and (3.9) and taking into account
the fact that fls = 0 for l < s ≤ m and fsj = 0 for s ≥ m + 1, we have
j−1
(i)
flj (tll

−

(i)
tjj )

=

(i)
tlj

(i)

(i)

(fls tsj − tls fsj )

+
s=l+1

j−1

m

=

(i)
tlj

(i)
(fls tsj

+

−

(i)
tls fsj )

s=m+1

s=l+1
j−1

m

=

(i)
tlj

(i)
tls fsj

−

(i)

(i)

+

fls tsj
s=m+1

s=l+1

(1 ≤ i ≤ m, m + 1 ≤ j ≤ N; 1 ≤ i ≤ k).

11

(i)

(fls tsj − tls fsj )

+

(3.10)


Since
(i)

(i)

flj = xl,j−m , tlj = cl,j−m for all 1 ≤ l ≤ m, m + 1 ≤ j ≤ N.
tsj = asj for all 1 ≤ s, j ≤ m, ts+m,j+m = bsj for all 1 ≤ s, j ≤ n,
(3.10) implies the following relations for the elements xlj of X:
j−1

m
(i)
xlj (all

−

(i)
bjj )

=

(i)
clj

(i)
als xsj

−

(i)

+

xls bsj ,

(3.11)

s=1

s=l+1

for all i, j and l (1 ≤ i ≤ k, 1 ≤ j ≤ n, 1 ≤ l ≤ m). Observe that the condition
σ(A1, ..., Ak) ∩ σ(B1 , ..., Bk) = ∅ implies that
For all 1 ≤ l ≤ m, 1 ≤ j ≤ n, there exists p := p(l, j), 1 ≤ p ≤ k
(p)

(p)

such that all = bjj ,

(3.12)

so that
xlj =

1
(p)
all

−

j−1

m
(p)
[c
(p) lj
bjj

(p)
als xsj

−
s=l+1

(p)

xls bsj ], (1 ≤ l ≤ m, 1 ≤ j ≤ n). (3.13)

+
s=1

Relations (3.13) allow us to compute elements xlj of the solution matrix X recursively,
one sub diagonal, middle diagonal or super diagonal at a time, starting from the
element xm1 , in the following order:
xm1
xm−1,1 , xm2
xm−2,1 , xm−1,2, xm−2,3
·········
Remark 3.3. As in the case of functions of several commuting matrices (see Remark
3.1), each value xlj must satisfy the recurrence relations (3.11) for all i = 1, ..., k, but
is computed using one value p such that (3.12) holds. However, it is consistent in the
sense that the relations (3.11) for all other values i (1 ≤ i ≤ k) are automatically
satisfied.
Thus, we obtain the following algorithm for finding the simultaneous solution X of
equations (1.2). This algorithm consists, in general, of three parts, corresponding to
finding elements xlj on sub diagonals, middle diagonals and super diagonals, respectively. Assume, for definiteness, that m ≤ n (the corresponding algorithm for the
case m > n is analogous).
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Part 1. For finding
xm1 ;
xm−1,1 , xm,2 ;
···
x11 , ..., xmm.
For each l = 0, ..., m − 1
for each j = 1, ..., l + 1
(p)

(p)
find p such that am−l+j−1,m−l+j−1 = bjj and compute
xm−l+j−1,j =

1
(p)

j−1

m
(p)

(p)

[c
−
(p) m−l+j−1,j

am−l+j−1,m−l+j−1 − bjj

(p)

am−l+j−1,s xsj +

xm−l+j−1,s bsj ].
s=1

s=m−l+j

Part 2. For finding

x12 , x23 , ..., xm,m+1;
x13 , x24 , ..., xm,m+2;
···
x1,n−m+1 , x2,n−m+2 , ..., xm,n.
For each j = 0, ..., n − m − 1
for each l = 1, ..., m,
(p)
(p)
find p such that all = bl+j+1,l+j+1 and compute
xl,l+j+1 =

(p)
1
[c
−
(p)
(p)
all −bl+j+1,l+j+1 l,l+j+1
l+j (p)
s=1 bs,l+j+1 ].

m
s=l+1

(p)

als xs,l+j+1 +

Part 3. For finding
x1,n−m+2 , x2,n−m+3 , ..., xm−1,n;

x1,n−m+3 , x2,n−m+4 , ..., xm−2,n;
···
x1,n−1, x2,n;
x1,n.
For each j = 1, ..., m − 1
for each l = 1, ..., m − j,
(p)
(p)
find p such that all = bn−m+j+l,n−m+j+l , and compute
(p)
1
[c
−
(p)
(p)
all −bn−m+j+l,n−m+j+l l,n−m+j+l
m
(p)
n−m+j+l−1
(p)
xls bs,n−m+j+l ]
s=l+1 als xs,l,n−m+j+l +
s=1

xl,n−m+j+l =

Observe that we only use the commutativity conditions of the k-tuples (A1, ..., Ak)
and (B1, ..., Bk ) to ensure that the solution matrix X exists. Suppose now that
(A1, ..., Ak) and (B1 , ..., Bk) are arbitrary, not necessarily commuting, k-tuples of
13

n × n and m × m upper triangular matrices, and (C1, ..., Ck ) is an arbitrary k-tuple
of m × n matrices. We can define the joint spectrum σ(A1, ..., Ak) by reading off the
(i)
diagonal elements as in the case of commuting matrices; namely, if Ai = (alj )m
l,j=1 ,
(1)

(k)

then σ(A1, ..., Ak) = {(aii , ..., aii ) : i = 1, ..., m}. Assume that σ(A1, ..., Ak) ∩
σ(B1, ..., Bk ) = ∅, and consider the system of Sylvester equations (1.2).
According to Proposition 2.1, a matrix X is a simultaneous solution of equations
(1.2) if and only if FX Ti = TiFX for all i = 1, ..., k, where FX and Ti are defined by
(1.4)-(1.5). The commutativity relations FX Ti = Ti FX , in turn, imply the relations
(3.11), and the condition σ(A1, ..., Ak) ∩ σ(B1, ..., Bk ) = ∅ implies that (3.12) holds.
But now it may occur that for certain values l, j (1 ≤ l ≤ m, 1 ≤ j ≤ n), the value
xlj which is calculated using the formula (3.13) for a certain value p such as (3.12)
holds, does not satisfy relations (3.11) for some other index i = p. In this case there
does not exist a simultaneous solution X.
Therefore, the above algorithm for finding simultaneous solutions X of equations
(1.2) also is applicable for arbitrary, not necessarily commuting, upper triangular
k-tuples (A1, ..., Ak ) and (B1 , ..., Bk), and for arbitrary k-tuple of m × n matrices
(C1 , ..., Ck), but with a modification. Namely, after each value xlj is computed using
the relation (3.13) for a particular value p such that (3.12) holds, one must check that
it also satisfies (3.11) for all other values i (1 ≤ i ≤ k). If (3.11) is not satisfied for
just one value i = p (1 ≤ i ≤ k), then the algorithm is aborted and the conclusion is
that there is no simultaneous solution of equations (1.2).
References

1. B. Bakkaloˇ
glu, K. Erciye¸s and C
¸ .K. Ko¸c, A parallelization of Parlett’s algorithm for functions
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2. R.H. Bartels and G.W. Stewart, Solution of the matrix equation AX + XB = C: Algorithm 432,
Commun. ACM 15(1972), 820-826.
3. R. Bhatia and P. Rosenthal, How and why to solve the operator equation AX − XB = Y ? Bull.
London Math. Soc. 29(1997), 1-21.
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SIAM Journal Matrix Anal. Appl. 25 (2)(2003), 464-485.
5. G.H. Golub, S. Nash and C.F. Van Loan, A Hessenberg-Schur method for the problem AX +
XB = C, IEEE Trans. Automat. Control AC-24(1979), 909-913.
6. J.J. Koliha, Block diagonalization, Mathematica Bohemica 126(2001), 237-246.
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10. J. Sylvester, Sur l’equations en matrices px = xq, C.R. Acad. Sci. Paris 99(1884), 67-71, 115-116.
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea
E-mail address:
Department of Mathematics, Ohio University, Athens 45701, USA, and, Vietnam
Institute of Advanced Study in Mathematics, Hanoi, Vietnam
E-mail address:

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