Expressing infinite matrices as sums of idempotents
Let $\scr M_{Cf} (F)$ be the set of all column-finite $N \times N$ matrices over a field $F$. The following problem is studied: what elements of $\scr M_{Cf} (F)$ can be expressed as a sum of idempotents? The result states that every element of $\scr M_{Cf} (F)$ can be represented as the sum of 14...
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| author | Słowik, R. Словік, Р. |
| author_facet | Słowik, R. Словік, Р. |
| author_sort | Słowik, R. |
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| description | Let $\scr M_{Cf} (F)$ be the set of all column-finite $N \times N$ matrices over a field $F$. The following problem is studied: what
elements of $\scr M_{Cf} (F)$ can be expressed as a sum of idempotents? The result states that every element of $\scr M_{Cf} (F)$ can
be represented as the sum of 14 idempotents. |
| first_indexed | 2026-03-24T02:12:14Z |
| format | Article |
| fulltext |
UDC 512.5
R. Słowik (Inst. Math., Silesian Univ. Technology, Gliwice, Poland)
EXPRESSING INFINITE MATRICES AS SUMS OF IDEMPOTENTS
ЗОБРАЖЕННЯ НЕСКIНЧЕННИХ МАТРИЦЬ
У ВИГЛЯДI СУМ IДЕМПОТЕНТIВ
Let \scrM Cf (F ) be the set of all column-finite \BbbN \times \BbbN matrices over a field F. The following problem is studied: what
elements of \scrM Cf (F ) can be expressed as a sum of idempotents? The result states that every element of \scrM Cf (F ) can
be represented as the sum of 14 idempotents.
Нехай \scrM Cf (F ) — множина всiх \BbbN \times \BbbN матриць зi скiнченними стовпчиками над полем F. Вивчається наступна
проблема: якi елементи \scrM Cf (F ) можна зобразити у виглядi суми iдемпотентiв? Показано, що кожний елемент
\scrM Cf (F ) можна зобразити у виглядi суми 14 iдемпотентiв.
1. Introduction. It is a classical question whether the elements of a ring or a group can be expressed
as sums or products of elements of some particular set. One of the most known problems is expressing
the elements as a sum of a unit and an idempotent [1, 3, 5]. Rings in which every element can be
written in such way are called clean and have a special place in ring theory [4]. However, even more
often we are interested in situation when the elements can be written as sums of the elements sharing
some property. One example of such property is being square-zero [6, 22]. Another example, that will
be of our interest in this paper, is idempotency. The first result in this field is due to Stampfli [20] who
proved that any bounded linear operator on a Hilbert space is a sum of at most 8 projections. Further
research in this direction showed that in some cases this number can be even less (see [12, 14]).
In [7] Hartwig and Putcha considered the matrices over a field \BbbF and posed the following
questions:
(Qf1) When is A an \BbbF -linear combination of idempotents?
(Qf2) When is A a \pm 1-combination of idempotents?
(Qf3) When is A a sum of idempotents?
(Qf4) When is A a positive linear combination of idempotents (\BbbF = \BbbR )?
In particular they answered (Qf3) and proved the following theorem.
Theorem 1.1 ([7], Theorem 1). Let M \in \BbbF n\times n. Then M is a sum of idempotents if and only if
\mathrm{t}\mathrm{r}(M) = ke, where k \in \BbbZ and k \geq \rho (M).
(The symbol \rho stands here for the rank and \mathrm{t}\mathrm{r} stands for the trace.)
Let us note that the same result was also proved by Wu in [23] who considered also the minimal
number of required idempotents. The problem of finding minimal number was studied further in
[21]. In particular, it is of interest how we can describe matrices that are sums of some fixed, usually
quite small, number of idempotents [10]. We should also mention that from [8] we know the form
of any commutative ring in which every element is a sum of two idempotents and from [9] we know
the form of any algebra generated by two idempotents.
As we can easily observe not every matrix is a sum of idempotents. Because of that the answer
to question (Qf1) seems to be very interesting as well. In [13] it is proved that if characteristic of
the field is equal to 0, then every matrix is a linear combination of at most 3 idempotents. It is also
known that for the operators acting on a Hilbert space we need 5 projections [11]. Sometimes this
number can be even smaller [15].
c\bigcirc R. SŁOWIK, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8 1145
1146 R. SŁOWIK
All the result we have mentioned so far hold for the matrices over fields with characteristic 0.
This problem for the fields of positive characteristic was solved only in 2010 by de Seguins Pazzis
[18]. Let us present the solution.
Theorem 1.2 ([18], Theorem 4). A matrix A \in Mn(\BbbK ) is a sum of idempotents iff \mathrm{t}\mathrm{r}A \in \BbbF p.
In particular, every matrix of Mn(\BbbF p) is a sum of idempotents.
This author found also the form of the matrices that are linear combinations of 2 and of 3
idempotents over an arbitrary field [16, 17].
In the present paper we would like to study the problem whether a \BbbN \times \BbbN matrix can be written as
a sum of idempotents. We will consider the matrices with the property that each column contains only
a finite number of nonzero entries and we will call them column-finite. The set of all column-finite
matrices over a field F will be denoted by \scrM Cf (F ).
Our main result is the following theorem.
Theorem 1.3. Assume that F is a field. Any matrix a \in \scrM Cf (F ) can be expressed as a sum of
at most 14 idempotents \scrM Cf (F ).
2. Proofs. We start with introducing the notation.
We write e\infty and ek for \BbbN \times \BbbN and k\times k identity matrices, respectively, and enm for the \BbbN \times \BbbN
matrix with 1 in the position (n,m) and 0 in every other position.
By \scrT \infty (F ) we denote the subring of \scrM Cf (F ) consisting of all upper triangular matrices, and
by \scrL TCf (F ) — the subring of all column-finite lower triangular matrices. The symbols \scrT n(F ) and
\scrL Tn(F ) will be used for the rings of all n \times n upper or lower triangular matrices respectively,
whereas \scrD n(F ) will denote the ring of all n\times n diagonal matrices. The full n\times n matrix ring will
be denoted by \scrM n\times n(F ).
If a is any matrix and b is invertible, then by ab we mean b - 1ab.
If we write that an integer k \in F then by k we mean the element
1 + 1 + . . .+ 1\underbrace{} \underbrace{}
k
,
where 1 is the identity in F. By \langle 1\rangle we will denote a subring of F that is generated by 1. The symbol
F \ast stands for F \setminus \{ 0\} .
If an infinite matrix a has a nonzero entries only in the positions (i, j) from some set I, then we
will write
a =
\sum
(i,j)\in I
aijeij .
Note that in this context \Sigma is a notation, not a traditional sum.
In the whole paper we will use without a reference the following.
Remark 2.1. The matrix a \in \scrM Cf (F ) is a sum/linear combination of idempotents if and only
if any b similar to a (in \scrM Cf (F )) is a sum/linear combination of idempotents.
2.1. Triangular matrices. Obviously we would like to deal with matrices of quite simple
structure, for instance containing as many zeros as possible. In order to do that we start with a lemma
that will help us with replacing the matrices by their conjugacies.
Lemma 2.1. Let F be a field and t \in \scrT \infty (F ). If t = e\infty +
\sum
m - n\geq 1
tnmenm and for all n \in \BbbN
we have tn,n+1 \not = 0, then t is similar to e\infty +
\sum \infty
n=1
tn,n+1en,n+1.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
EXPRESSING INFINITE MATRICES AS SUMS OF IDEMPOTENTS 1147
Proof. We will show that it is possible to find an invertible x \in \scrT \infty (F ) such that
tx = x
\Biggl(
e\infty +
\infty \sum
n=1
tn,n+1en,n+1
\Biggr)
.
From the above equation we conclude that the entries of x satisfy the system
tn,n+1xn+1,m +
m\sum
r=n+2
tnrxrm = xn,m - 1tm - 1,m for all m - n \geq 1. (1)
It can be solved, for instance, as below.
In the first step we substitute m = n+ 1 to (1) and get
tn,n+1xn+1,n+1 = xnntn,n+1.
This means that we can choose x11 arbitrarily from F \ast and for n \geq 2 we have xnn = x11.
In the second step we substitute m = n+ 2 to (1) and get
tn,n+1xn+1,n+2 + tn,n+2xn+2,n+2 = xn,n+1tn+1,n+2.
Since tn,n+1 \not = 0 we have
xn+1,n+2 = t - 1
n,n+1(xn,n+1tn+1,n+2 - tn,n+2xn+2,n+2). (2)
Hence, we can choose x12 arbitrarily from F and find the next entries from the first diagonal using (2).
In the sth step we substitute m = n+ 1 + s to (1) and have
tn,n+1xn+1,n+1+s + tn,n+2xn+2,n+1+s + tn,n+3xn+3,n+1+s + . . .+ tn,n+1+sxn+1+s,n+1+s =
= xn,n+stn+s,n+1+s.
Again we set x1,1+s arbitrarily and find the next elements of the sth diagonal using
xn+1,n+1+s = t - 1
n,n+1
\bigl(
xn,n+stn+s,n+1+s -
- tn,n+2xn+2,n+1+s - tn,n+3xn+3,n+1+s - . . . - tn,n+1+sxn+1+s,n+1+s
\bigr)
.
Performing this way we find all diagonals of x and consequently x itself.
We will use Lemma 2.1 to prove the following proposition.
Proposition 2.1. Let F be a field. If t \in \scrT \infty (F ) satsifies the condition tnn = 2 for all n \in \BbbN ,
then t is a sum of at most 4 idempotents.
Proof. First we define two matrices t1 and t2 as follows:
(t1)nm =
\left\{
1 if m = n,
tnm if m - n > 1,
tnm if m - n = 1 and tnm \not = 0,
1 if m - n = 1 and tnm = 0,
t2 = t - t1.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
1148 R. SŁOWIK
One can see that t1 fulfills the assumptions of Lemma 2.1. Hence, for some x we have
tx1 = e\infty +
\infty \sum
n=1
(t1)n,n+1en,n+1.
We write tx1 as u+ u\prime , where
u =
\infty \sum
n=1
\bigl(
e2n - 1,2n - 1 + (t1)2n - 1,2ne2n - 1,2n
\bigr)
, u\prime =
\infty \sum
n=1
\bigl(
e2n,2n + (t1)2n,2n+1e2n,2n+1
\bigr)
.
It is easy to check that u and u\prime are idempotents.
We deal similarly with t2. We write it as v\prime + v\prime \prime where v\prime is defined by the following inductive
rule:
(1) v\prime 11 = (t2)11, v
\prime
12 = (t2)12;
(2) if v\prime n,n+1 = 0, then put we v\prime n+1,n+1 = 1, v\prime n+1,n+2 = (t2)n+1,n+2;
(3) if v\prime n,n+1 \not = 0, then we put v\prime n+1,n+1 = 0, v\prime n+1,n+2 = 0.
The idea of this decomposition is depicted in Fig. 1.
Fig. 1. Picture to the proof of Proposition 2.1. If v\prime n,n+1 = \alpha \not = 0, then the entries from the next row ‘go’ to v\prime \prime . If
v\prime n,n+1 = 0, then the next row can ‘stay’ in v\prime .
From the construction of v\prime and v\prime \prime it follows that they are idempotents. Thus, the result follows.
Now we will focus on lower triangular matrices. Note that we consider only column-finite lower
triangular matrices. This restriction may cause some difficulties. In particular we can not make use
of Lemma 2.1, because since t is lower triangular, the matrix x may turn out not to be column-finite.
Therefore, we will first decompose t to a sum of two block matrices and then we will focus on those
two. In particular we will need some more information about finite matrices. It can be noticed that
from the method of proof of Proposition 2.1 we can derive the following corollary.
Corollary 2.1. Let F be a field and k \in \BbbN . If a is either from \scrT k(F ) or \scrL Tk(F ) and ann = 2
for all n, 1 \leq n \leq k, then a is a sum of at most 4 idempotents.
The above corollary will be useful in the proof of the below result.
Proposition 2.2. If F is a field and t \in \scrL TCf (F ) is such that tnn = 3 for all n \in \BbbN , then t
can be written as a sum of at most 6 idempotents.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
EXPRESSING INFINITE MATRICES AS SUMS OF IDEMPOTENTS 1149
Proof. We apply the method used in the proof of Proposition 2.2 from [19].
First we construct the sequence (lm)m\in \BbbN as follows:
lm = \mathrm{m}\mathrm{a}\mathrm{x} \{ i : tim \not = 0\} .
Now we define another sequence (l\prime m)m\in \BbbN as follows:
l\prime m =
\left\{ lm if for all j < m we have lj \leq lm,
\mathrm{m}\mathrm{a}\mathrm{x} \{ lj : j < m \wedge lj > lm\} otherwise.
The sequence (l\prime m)m\in \BbbN is nondecreasing. It can be observed that for all m \in \BbbN and all i > l\prime m we
have tim = 0. Thus, we can say that t has staircase structure and that the ‘stairs’ are determined by
the sequence (l\prime m). (See the ‘stairs’ in Fig. 2.)
Fig. 2. Picture to the proof of Proposition 2.2. The polygon arc depicts the staircase structure of t. It the red area there
are the nonzero coefficients of u, whereas in the blue area we have the nonzero coefficients of v.
Now we will write t as a sum of two matrices: u and v. The matrix u is constructed as follows.
For all i \in \BbbN we put uii = 2. Now let m be equal to 1 and let n be equal to l\prime 1. For all i, j such
that m \leq j < i \leq n we put uij = tij .
Let now m be equal to the preceding n increased by 1 and let the new n be equal to l\prime m. Then
for all i, j, m \leq i, j \leq n we put uij = tij .
Proceeding the same way we obtain a block diagonal matrix u (in Fig. 2 it is marked by red
color). Clearly v = t - u.
By Corollary 2.1 each block of u is a sum of at most 4 idempotents. Thus, so is u. From the
construction of v it follows that
v =
\left(
ek1
v1 ek2
0 v2 ek3
0 0 v3 ek4
0 0 0 v4 ek5
...
...
. . . . . .
\right)
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
1150 R. SŁOWIK
We put
v\prime =
\left(
ek1
v1 0
0 0 ek3
0 0 v3 0
0 0 0 0 ek5
...
...
. . . . . .
\right)
,
v\prime \prime = v - v\prime =
\left(
0
0 ek2
0 v2 0
0 0 0 ek4
0 0 0 v4 0
...
...
. . . . . .
\right)
.
It is easy to check that v\prime , v\prime \prime are idempotents.
Proposition 2.2 is proved.
2.2. Diagonal matrices. We start with an observation about 2 \times 2 matrices that follows from
the proof of Theorem 1.1.
Lemma 2.2. If x \in F, where F is a field, then the matrix\Biggl(
x 0
0 2 - x
\Biggr)
is a sum of two idempotents.
Now we can give proof of the result that holds for \scrD \infty (F ).
Proposition 2.3. Let F be any field. Any d \in \scrD \infty (F ) is a sum of 4 idempotents from \scrM Cf (F ).
Proof. We write d as a sum of x, y \in \scrD \infty (F ), where the entries of x, y are defined inductively
according to the scheme as below:
x11 = d11, y11 = 0,
x22 = 2 - x11, y22 = d22 - x22,
y33 = 2 - y22, x33 = d33 - y33,
x44 = 2 - x33, y44 = d44 - x44,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Then x and y are of the forms:
x =
\left(
x11 0
0 2 - x11
x33 0
0 2 - x33
. . .
\right) ,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
EXPRESSING INFINITE MATRICES AS SUMS OF IDEMPOTENTS 1151
y =
\left(
0
y22 0
0 2 - y22
y44 0
0 2 - y44
. . .
\right)
,
and by Lemma 2.2 both x and y are sums of two idempotents.
Proposition 2.3 is proved.
2.3. Consequences. In this section we join the results from the two preceding ones and prove
our main theorem.
Proof of Theorem 1.3. Define t1, t2 and d as follows:
(t1)ij =
\left\{
2 if j = i,
aij if i < j,
0 if i > j,
(t2)ij =
\left\{
3 if j = i,
aij if i > j,
0 if i < j,
dij =
\left\{ aij - 5 if j = i,
0 if i \not = j.
Clearly a = t1 + t2 + d. From Propositions 2.1 and 2.2 we know that t1 and t2 are sums of at most
4 and 6 idempotents, respectively. By Proposition 2.3 the matrix d is a sum of 4 idempotents. This
proves the claim.
References
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1152 R. SŁOWIK
17. de Seguins Pazzis C. On decomposing any matrix as a linear combination of three idempotents // Linear Algebra and
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Received 22.10.15,
after revision — 08.04.17
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 8
|
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| spelling | umjimathkievua-article-17652019-12-05T09:25:58Z Expressing infinite matrices as sums of idempotents Зображення нескiнченних матриць у виглядi сум iдемпотентiв Słowik, R. Словік, Р. Let $\scr M_{Cf} (F)$ be the set of all column-finite $N \times N$ matrices over a field $F$. The following problem is studied: what elements of $\scr M_{Cf} (F)$ can be expressed as a sum of idempotents? The result states that every element of $\scr M_{Cf} (F)$ can be represented as the sum of 14 idempotents. Нехай $\scr M_{Cf} (F)$ — множина всiх $N \times N$ матриць зi скiнченними стовпчиками над полем $F$. Вивчається наступна проблема: якi елементи $\scr M_{Cf} (F)$ можна зобразити у виглядi суми iдемпотентiв? Показано, що кожний елемент $\scr M_{Cf} (F)$ можна зобразити у виглядi суми 14 iдемпотентiв. Institute of Mathematics, NAS of Ukraine 2017-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1765 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 8 (2017); 1145-1147 Український математичний журнал; Том 69 № 8 (2017); 1145-1147 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1765/747 Copyright (c) 2017 Słowik R. |
| spellingShingle | Słowik, R. Словік, Р. Expressing infinite matrices as sums of idempotents |
| title | Expressing infinite matrices as sums of idempotents |
| title_alt | Зображення нескiнченних матриць
у виглядi сум iдемпотентiв |
| title_full | Expressing infinite matrices as sums of idempotents |
| title_fullStr | Expressing infinite matrices as sums of idempotents |
| title_full_unstemmed | Expressing infinite matrices as sums of idempotents |
| title_short | Expressing infinite matrices as sums of idempotents |
| title_sort | expressing infinite matrices as sums of idempotents |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1765 |
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