On linear combinations of two tripotent, idempotent, and involutive 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.…”

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

“…It is a smart idea to use method (M2) on the corresponding problem without commutativity between the two special matrices; but it seems to bring lots of difficulties for the problems listed in (Q1)-(Q4), because it is not easy to find the proper matrices to multiply laterally to make the method (M2) work effectively. However, if we can utilize method (M1) to tackle with similar problems, we may avoid the difficulty of finding the right way of multiplying a proper matrix laterally on both sides of a matrix equation, as demonstrated in [19].…”

“…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].…”

On the spectra of some combinations of two generalized quadratic matrices

“…In [8], Sarduvan andÖzdemir proved that for c 1 , c 2 ∈ * and idempotent matrices T 1 , T 2 ∈ n , under the assumption T 1 T 2 = T 2 T 1 , the matrix c 1 T 1 + c 2 T 2 is involutive if c 1 + c 2 = 0 and 1…”