Commutativity of projectors

A family of 512 reverse order laws for generalized inverses of a matrix product: A review

“…For example, the basic formulas in the following lemma can be found, e.g., in [15,17,65]. The general expressions of the seven commonly-used types of generalized inverses (1,3,4) , (1,2,4) , (1,2,3) , (1,4) , (1,3) , (1,2) , and (1) of can be written in the following 7 matrix-valued functions (1,3,4…”

“…If ( ) = , the general expressions of the last seven generalized inverses of in (1.2) can be written as (1,3,4) = (1,2,4) = (1,4…”

“…(c) The following set inclusions hold † ∈ { (1,3,4) } ⊆ { (1,4) } ⊆ { (1) }, † ∈ { (1,3,4) } ⊆ { (1,3) } ⊆ { (1) }, (3.26) † ∈ { (1,2,4) } ⊆ { (1,4) } ⊆ { (1) }, † ∈ { (1,2,4) } ⊆ { (1,2) } ⊆ { (1) }, (3.27) † ∈ { (1,2,3) } ⊆ { (1,3) } ⊆ { (1) }, † ∈ { (1,2,3) } ⊆ { (1,2) } ⊆ { (1) }, (3.28) and following matrix set equalities hold {( (1,3,4) ) * } = {( * ) (1,3,4) }, {( (1,2,4) ) * } = {( * ) (1,2,3) }, (3.29) {( (1,2,3) ) * } = {( * ) (1,2,4) }, {( (1...…”

“…The collection of all { , … , }-generalized inverses of is denoted by { ( ,…, ) }. There are all 15 types of { , … , }-generalized inverses of by definition, but people are mainly interested in the following eight situations that involve the equation = : † , (1,3,4) , (1,2,4) , (1,2,3) , (1,4) , (1,3) , (1,2) , (1) , (1.2) which are usually called the eight commonly-used types of generalized inverses of in the literature (cf. [15,17,65]).…”

On the idempotency, involution and nilpotency of a linear combination of two matrices

“…In Tables 1 and 2 Baksalary characterized all situations in which a linear combination of two idempotent matrices is idempotent, which corresponds to cases (1) and (5) in Table 1. [2] For the remaining solved problems, the readers are referred to [2,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. For recent advances on this topic and related papers, see [30][31][32][33][34][35][36][37][38][39][40][41][42].…”