‎In this paper‎, ‎some algebraic and geometrical properties of the rank$-k$ numerical hulls of normal matrices are investigated‎. ‎A characterization of normal matrices whose rank$-1$ numerical hulls are equal to their numerical range is given‎. ‎Moreover‎, ‎using the extreme points of the numerical range‎, ‎the higher rank numerical hulls of matrices of the form $A_1 oplus i A_2$‎, ‎where $A_1...

‎A rank equality is established for the sum of finitely many tripotent matrices via elementary block matrix operations‎. ‎Moreover‎, ‎by using this equality and Theorems 8 and 10 in [Chen M‎. ‎and et al‎. ‎On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications‎, ‎The Scientific World Journal 2014 (2014)‎, ‎Article ID 702413‎, ‎7 page...

Motivated by (and using tools from) communication complexity, we investigate the relationship between the following two ranks of a 0-1 matrix: its nonnegative rank and its binary rank (the log of the latter being the unambiguous nondeterministic communication complexity). We prove that for partial 0-1 matrices, there can be an exponential separation. For total 0-1 matrices, we show that if the ...

For a rank-1 matrix A = a ⊗ b over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T (A) = U ⊗ A ⊗ V , or T (A) = U ⊗ A ⊗ V with ...