SEVERAL QUESTIONS OF LONGSTAFF ON GENERATION IN MATRIX ALGEBRAS

Source abstract

Abstract We solve three problems raised in recent articles by Longstaff [ Bull. Aust. Math. Soc. 102 (2020), 226–236; 103 (2021), 260–270; 104 (2021), 78–93]. We show that: (1) the length of an irreducible family of n × n $n\times n$ n times n rank-one matrices can take any value 1 , 2 , … , n $1,2,\ldots ,n$ 1 comma 2 comma ellipsis comma n ; (2) the slot length of an irreducible pair of n × n $n\times n$ n times n matrices can exceed 2 n − 4 $2n-4$ 2 n minus 4 ; and (3) we give bounds on the 1 $1$ 1 -minimum spanning lengths of irreducible pairs.