Scalarmatrice

Scalarmatrice is a term used in Italian mathematics to refer to a scalar matrix, a special kind of diagonal matrix that represents a uniform scaling by a single scalar factor. Formally, a scalarmatrice of size n over a field F is a matrix of the form cI_n, where c ∈ F and I_n is the n×n identity matrix. In linear terms, it corresponds to the linear transformation that multiplies every vector by the same scalar c, so it effects isotropic scaling of the entire space F^n.

Properties and consequences: all eigenvalues of a scalarmatrice are equal to c, its determinant is c^n, and its trace is n·c. It is invertible precisely when c ≠ 0, with inverse (1/c)I_n. A scalarmatrice commutes with every n×n matrix, so the set of all scalar matrices forms the center of the matrix algebra M_n(F). In the general linear group GL(n, F), the scalar matrices form a subgroup {cI_n : c ∈ F^×}, isomorphic to the multiplicative group F^×.

Relation to other concepts: a scalarmatrice is a special case of a diagonal matrix, distinct from diagonal matrices with nonuniform diagonal entries. It is also referred to as a matrice scalare in some Italian texts. The concept is central in discussions of isotropic scaling and in characterizing linear operators that act by uniform scaling on the entire space.

See also: matrice scalare, identità, trasformazioni lineari, GL(n, F), centro di M_n(F).