Editing Diagonal matrix (section)

== Scalar matrix ==
{{Confusing|section|reason=many sentences use incorrect, awkward grammar and should be reworded to make sense|date=February 2021}}
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A diagonal matrix with equal diagonal entries is a '''scalar matrix'''; that is, a scalar multiple {{mvar|λ}} of the [[identity matrix]] {{math|'''I'''}}. Its effect on a [[vector (mathematics and physics)|vector]] is [[scalar multiplication]] by {{mvar|λ}}. For example, a 3×3 scalar matrix has the form:
<math display="block">
  \begin{bmatrix}
    \lambda &       0 & 0       \\
          0 & \lambda & 0       \\
          0 &       0 & \lambda
  \end{bmatrix} \equiv \lambda \boldsymbol{I}_3
</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 [[field (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>\mathbf{D} = \operatorname{diag}(a_1, \dots, a_n)</math> has <math>a_i \neq a_j,</math> then given a matrix {{math|'''M'''}} with <math>m_{ij} \neq 0,</math> the {{math|(''i'', ''j'')}} term of the products are: <math>(\mathbf{DM})_{ij} = a_im_{ij}</math> and <math>(\mathbf{MD})_{ij} = m_{ij}a_j,</math> and <math>a_jm_{ij} \neq m_{ij}a_i</math> (since one can divide by {{mvar|m{{sub|ij}}}}), 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 {{mvar|V}} (rather than the concrete vector space {{mvar|K{{sup|n}}}}), the analog of scalar matrices are '''scalar transformations'''. This is true more generally for a [[module (ring theory)|module]] {{mvar|M}} over a [[ring (algebra)|ring]] {{mvar|R}}, with the [[endomorphism algebra]] {{math|End(''M'')}} (algebra of linear operators on {{mvar|M}}) replacing the algebra of matrices. Formally, scalar multiplication is a linear map, inducing a map <math>R \to \operatorname{End}(M),</math> (from a scalar {{mvar|&lambda;}} to its corresponding scalar transformation, multiplication by {{mvar|&lambda;}}) exhibiting {{math|End(''M'')}} as a {{mvar|R}}-[[Algebra (ring theory)|algebra]]. For vector spaces, the scalar transforms are exactly the [[center of a ring|center]] of the endomorphism algebra, and, similarly, scalar invertible transforms are the center of the [[general linear group]] {{math|GL(''V'')}}. The former is more generally true [[free module]]s <math>M \cong R^n,</math> for which the endomorphism algebra is isomorphic to a matrix algebra.