|
</math>
The scalar matrices are the [[center of an algebra|center]] of the algebra of matrices: that is, they are precisely the matrices that [[commute (mathematics)|commute]] with all other square matrices of the same size.{{efn|Proof: given the [[elementary matrix]] <math>e_{ij}</math>, <math>Me_{ij}</math> is the matrix with only the ''i''-th row of ''M'' and <math>e_{ij}M</math> is the square matrix with only the ''M'' ''j''-th column, so the non-diagonal entries must be zero, and the ''i''th diagonal entry much equal the ''j''th diagonal entry.}} By contrast, over a [[Fieldfield (mathematics)|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 <math>D = \operatorname{diag}(a_1, \dots, a_n)</math> has <math>a_i \neq a_j,</math> then given a matrix <math>M</math> with <math>m_{ij} \neq 0,</math> the <math>(i, j)</math> term of the products are: <math>(DM)_{ij} = a_im_{ij}</math> and <math>(MD)_{ij} = m_{ij}a_j,</math> and <math>a_jm_{ij} \neq m_{ij}a_i</math> (since one can divide by <math>m_{ij}</math>), so they do not commute unless the off-diagonal terms are zero.{{efn|Over more general rings, this does not hold, because one cannot always divide.}} Diagonal matrices where the diagonal entries are not all equal or all distinct have centralizers intermediate between the whole space and only diagonal matrices.<ref>{{cite web |url=https://math.stackexchange.com/q/1697991 |title=Do Diagonal Matrices Always Commute? |author=<!--Not stated--> |date=March 15, 2016 |publisher=Stack Exchange |access-date=August 4, 2018 }}</ref>
For an abstract vector space ''V'' (rather than the concrete vector space <math>K^n</math>), orthe analog of scalar matrices are '''scalar transformations'''. This is true more generally for a [[module (ring theory)|module]] ''M'' over a [[ring (algebra)|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 <math>R \to \operatorname{End}(M),</math> (sendfrom a scalar ''λ'' to theits corresponding scalar transformation, multiplication by ''λ'') exhibiting End(''M'') as a ''R''-[[Algebra (ring theory)|algebra]]. For vector spaces, or more generally [[free module]]s <math>M \cong R^n</math>, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the [[center of a ring|center]] of the endomorphism algebra, and, similarly, invertible transforms are the center of the [[general linear group]] GL(''V''),. whereThe theyformer areis denotedmore bygenerally Z(''V''),true follow[[free themodule]]s usual<math>M notation\cong R^n</math>, for which the centerendomorphism algebra is isomorphic to a matrix algebra.
== Vector operations ==
|
