zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The nullity and rank of linear combinations of idempotent matrices. (English) Zbl 1104.15001
{\it J. K. Baksalary} and {\it O. M. Baksalary} [Linear Algebra Appl. 388, 25--29 (2004; Zbl 1081.15017)] have shown that if $P_1$, $P_2$ are idempotent matrices (i.e. $P_j^2=P_j$), then the nonsingularity of $P_1+P_2$ is equivalent to the one of any linear combination $P:=c_1P_1+c_2P_2$, $c_j\in {\Bbb C}^*$, $c_1+c_2\neq 0$. In the present note the authors show that the nullity (i.e. the dimension of the nullspace) and rank of $P$ are constant. They provide a simple proof of a rank formula from {\it J. Groß} and {\it G. Trenkler} [SIAM J. Matrix Anal. Appl. 21, 390--395 (1999; Zbl 0946.15020)].

15A03Vector spaces, linear dependence, rank
15A24Matrix equations and identities
Full Text: DOI
[1] Baksalary, J. K.; Baksalary, O. M.: Nonsingularity of linear combinations of idempotent matrices. Linear algebra appl. 388, 25-29 (2004) · Zbl 1081.15017
[2] Groß, J.; Trenkler, G.: Nonsingularity of the difference of two oblique projectors. SIAM J. Matrix anal. Appl. 21, 390-395 (1999) · Zbl 0946.15020
[3] Koliha, J. J.; Rakočević, V.; Straškraba, I.: The difference and sum of projectors. Linear algebra appl. 388, 279-288 (2004)
[4] Koliha, J. J.; Rakočević, V.: Fredholm properties of the difference of orthogonal projections in a Hilbert space. Integral equations operator theory 52, 125-134 (2005) · Zbl 1082.47009
[5] Marsaglia, G.; Styan, G. P. H.: Equalities and inequalities for ranks of matrices. Linear and multilinear algebra 2, 269-292 (1974) · Zbl 0297.15003