A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple λI of the identity matrix I. Its effect on a vector is scalar multiplication by λ. For example, a 3×3 scalar matrix has the form:
The scalar matrices are the center of the algebra of matrices: that is, they are precisely the matrices that commute with all other square matrices of the same size.
For an abstract vector space V (rather than the concrete vector space ), or more generally a module M over a ring R, with the endomorphism algebra End(M) (algebra of linear operators on M) replacing the algebra of matrices, the analog of scalar matrices are scalar transformations. Formally, scalar multiplication is a linear map, inducing a map (send a scalar λ to the corresponding scalar transformation, multiplication by λ) exhibiting End(M) as a R-algebra. For vector spaces, or more generally free modules, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the center of the endomorphism algebra, and similarly invertible transforms are the center of the general linear group GL(V), where they are denoted by Z(V), follow the usual notation for the center.
Read more about this topic: Diagonal Matrix
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