stylistic:style_transfer, linguistic:voice_change, structural:summarization, information:classification, structural:scholar_edition
Scalar matrix

A diagonal matrix with equal diagonal entries is a scalar matrix; that is, a scalar multiple λ 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. By contrast, over a field (like the real numbers), a diagonal matrix with all diagonal elements distinct only commutes with diagonal matrices (its centralizer is the set of diagonal matrices). That is because if a diagonal matrix D = diag (a 1, …, a n) {\displaystyle \mathbf {D} =\operatorname {diag} (a_{1},\dots,a_{n})} has a i ≠ a j, {\displaystyle a_{i}\neq a_{j},} then given a matrix M {\displaystyle \mathbf {M} } with m i j ≠ 0, {\displaystyle m_{ij}\neq 0,} the (i, j) {\displaystyle (i,j)} term of the products are: (D M) i j = a i m i j {\displaystyle (\mathbf {D} \mathbf {M})_{ij}=a_{i}m_{ij}} and (M D) i j = m i j a j, {\displaystyle (\mathbf {M} \mathbf {D})_{ij}=m_{ij}a_{j},} and a j m i j ≠ m i j a i {\displaystyle a_{j}m_{ij}\neq m_{ij}a_{i}} (since one can divide by m i j {\displaystyle m_{ij}}), so they do not commute unless the off-diagonal terms are zero. Diagonal matrices where the diagonal entries are not all equal or all distinct have centralizers intermediate between the whole space and only diagonal matrices.

For an abstract vector space V (rather than the concrete vector space K n {\displaystyle K^{n}}), the analog of scalar matrices are scalar transformations. This is true more generally for a module M over a ring R, with the endomorphism algebra End(M) (algebra of linear operators on M) replacing the algebra of matrices. Formally, scalar multiplication is a linear map, inducing a map R → End (M), {\displaystyle R\to \operatorname {End} (M),} (from a scalar λ to its corresponding scalar transformation, multiplication by λ) exhibiting End(M) as a R - algebra. For vector spaces, the scalar transforms are exactly the center of the endomorphism algebra, and, similarly, scalar invertible transforms are the center of the general linear group GL(V). The former is more generally true free modules M ≅ R n {\displaystyle M\cong R^{n}}, for which the endomorphism algebra is isomorphic to a matrix algebra.