On idempotency and tripotency of linear combinations of two commuting tripotent matrices

Tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute

“…For the remaining cases, the readers are referred to [1,3,4,[6][7][8][9][12][13][14]16,17]. In this paper, we solve cases (30) and (31), which are special cases of (32) and (33) corresponding to open problems as well.…”

A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications

“…This matrix pencil was considered in [4,11,12]. From (17), the matrix M in (52) can be rewritten in the following eight-term DID…”

“…The details are also omitted. Some previous work on idempotency and tripotency of linear combinations of two commuting tripotent matrices can be found in [4,5,11].…”

“…Finally, we consider a special case M ¼ a 1 A þ b 1 B of (2). This matrix pencil was studied by some authors in recent years; see, e.g., [4,5,[10][11][12]. Idempotent and tripotent matrices are two types of simpler matrix in the classification of matrices, which play a significant role in studying various problems in matrix theory and applications.…”

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].…”