Linear operators on matrices: the invariance of rank-k matrices

Commuting maps on rank-k matrices

On higher rank numerical hulls of normal matrices

‎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 brief introduction to quaternion matrices and linear algebra and on bounded groups of quaternion matrices

The division algebra of real quaternions, as the only noncommutative normed division real algebra up to isomorphism of normed algebras, is of great importance. In this note, first we present a brief introduction to quaternion matrices and quaternion linear algebra. This, among other things, will help us present the counterpart of a theorem of Herman Auerbach in the setting of quaternions. More ...

Commutative/noncommutative Rank of Linear Matrices and Subspaces of Matrices of Low Rank

A space of matrix of low rank is a vector space of rectangular matrices whose maximum rank is stricly smaller than the number of rows and the numbers of columns. Among these are the compression spaces, where the rank condition is garanteed by a rectangular hole of 0’s of appropriate size. Spaces of matrices are naturally encoded by linear matrices. The latter have a double existence: over the r...

On Linear Spaces of Skew-symmetric Matrices of Constant Rank

Linear sections of the Grassmannians G(1, n) of lines in P appear naturally in several different situations. In complex projective algebraic geometry, 3-dimensional linear sections of G(1, 4) appear in the classification of Fano threefolds, 2-dimensional linear sections of G(1, 5) define one of the smooth scrolls of P. Linear sections of dimension n − 1 of the Grassmannian of lines of P are cla...