Generalized and hypergeneralized projectors

On k-potent matrices

“…Furthermore, a number of papers shed light on the role which those matrices play in the problems occurring in the distribution of quadratic forms, e.g., [1], [2], [4], [6], [11], [12], [29], [37], and [38] to mention just a few. Quadripotent matrices recently focused also some special interest, which originates mostly from the fact that they occur naturally in considerations dealing with generalized and hypergeneralized projectors introduced in [20]. In addition to the papers [25], [27], [30], [31], and [43], each of which contains a systematical study over a selected topic concerning k-potent matrices, a collection of related isolated results was published in recent years in a number of independent articles.…”

Characterizations and the Moore-Penrose Inverse of Hypergeneralized K-Projectors

“…If a matrix A ∈ C n×n is similar to a diagonal matrix, then A is said to be diagonalizable. The concepts of g-p and hg-p were introduced by Groß and Trenkler [11] who presented very interesting properties of the classes of g-p and hg-p. A characterization of nonnegative matrices such that A = A † is derived by Berman [7].…”

“…This leads our interest to the subset of the class of square matrices A with the property A k = A † for k ∈ N and k > 1, called as hypergeneralized k-projectors. Specially, if k = 2, we get the class of h-p (see [1], [2], [3], [11], [13], [14]). …”

On linear combinations of two commuting hypergeneralized projectors

“…For instance, according to part (a) ⇔ (d) of Theorem 2 in [1] C HGP n = C EP n ∩ C QP n ; (1.5) see also Theorem 3 in [2]. A challenging and relevant question concerning matrices belonging to C HGP n is: when does a linear combination of the form (1.6) with c 1 , c 2 ∈ C and H 1 , H 2 ∈ C HGP n , inherit the hypergenerality property?…”