The c -spectral c -radial and c -convex matrices

Let c=(c 1,…cn ) be a complex row vector and [c] be the diagonal matrix with c 1,…cn as its diagonal entries. Given an n×n complex matrix A with eigenvalues α j , 1≦j≦n, we define as the c-eigenpolygonc-numerical rangec-spectral radiusc-numerical radius and c-spectral norm of A respectively. For c = (1,0,…, 0) they are reduced to the classical eigenpolygon, numerical range, spectral radius, numerical radius and spectral norm of A. We say that the matrix A is c-spectral if pc (A) =rc (A)c-radial if pc (A) = ||A|| c , and c-convex if Pc (A) = Wc (A). In this note we give characterizations of these matrices and study their properties.