Editing Matrix ring (section)

== Examples ==
* The set of all {{nowrap|''n'' × ''n''}} [[square matrices]] over ''R'', denoted M<sub>''n''</sub>(''R'').  This is sometimes called the "full ring of ''n''-by-''n'' matrices".
* The set of all upper [[triangular matrices]] over ''R''.
* The set of all lower [[triangular matrices]] over ''R''.
* The set of all [[diagonal matrices]] over ''R''.  This [[subalgebra]] of M<sub>''n''</sub>(''R'') is [[algebra homomorphism|isomorphic]] to the [[product of rings|direct product]] of ''n'' copies of ''R''.
* For any index set ''I'', the ring of endomorphisms of the right ''R''-module <math display="inline">M=\bigoplus_{i\in I}R</math> is isomorphic to the ring <math>\mathbb{CFM}_I(R)</math>{{fact|date=December 2020}}<!--Reference for this notation?--> of '''column finite matrices''' whose entries are indexed by {{nowrap|''I'' × ''I''}} and whose columns each contain only finitely many nonzero entries.  The ring of endomorphisms of ''M'' considered as a left ''R''-module is isomorphic to the ring <math>\mathbb{RFM}_I(R)</math> of '''row finite matrices'''.
* If ''R'' is a [[Banach algebra]], then the condition of row or column finiteness in the previous point can be relaxed.  With the norm in place, [[absolutely convergent series]] can be used instead of finite sums.  For example, the matrices whose column sums are absolutely convergent sequences form a ring.{{dubious|date=December 2020}}<!--Seems wrong, without any control on the growth rate in a row.-->  Analogously of course, the matrices whose row sums are absolutely convergent series also form a ring.{{dubious|date=December 2020}}  This idea can be used to represent [[Hilbert space#Operators on Hilbert spaces|operators on Hilbert spaces]], for example.
* The intersection of the row-finite and column-finite matrix rings forms a ring <math>\mathbb{RCFM}_I(R)</math>.
* If ''R'' is [[commutative ring|commutative]], then M<sub>''n''</sub>(''R'') has a structure of a [[*-algebra]] over ''R'', where the [[involution (mathematics)#Ring theory|involution]] * on M<sub>''n''</sub>(''R'') is [[matrix transpose|matrix transposition]]. 
* If ''A'' is a [[C*-algebra]], then M<sub>''n''</sub>(''A'') is another C*-algebra. If ''A'' is non-unital, then M<sub>''n''</sub>(''A'') is also non-unital. By the [[Gelfand–Naimark theorem]], there exists a [[Hilbert space]] ''H'' and an isometric *-isomorphism from ''A'' to a norm-closed subalgebra of the algebra ''B''(''H'') of continuous operators; this identifies M<sub>''n''</sub>(''A'') with a subalgebra of ''B''(''H''<sup>⊕''n''</sup>). For simplicity, if we further suppose that ''H'' is separable and ''A'' <math>\subseteq</math> ''B''(''H'') is a unital C*-algebra, we can break up ''A'' into a matrix ring over a smaller C*-algebra. One can do so by fixing a [[Projection (linear_algebra)#Orthogonal projections|projection]] ''p'' and hence its orthogonal projection 1&nbsp;−&nbsp;''p''; one can identify ''A'' with <math display="inline">\begin{pmatrix} pAp & pA(1-p) \\ (1-p)Ap & (1-p)A(1-p) \end{pmatrix}</math>, where matrix multiplication works as intended because of the orthogonality of the projections. In order to identify ''A'' with a matrix ring over a C*-algebra, we require that ''p'' and 1&nbsp;−&nbsp;''p'' have the same "rank"; more precisely, we need that ''p'' and 1&nbsp;−&nbsp;''p'' are Murray–von&nbsp;Neumann equivalent, i.e., there exists a [[partial isometry]] ''u'' such that {{nowrap|1=''p'' = ''uu''*}} and {{nowrap|1=1 − ''p'' = ''u''*''u''}}. One can easily generalize this to matrices of larger sizes.
* Complex matrix algebras M<sub>''n''</sub>('''C''') are, up to isomorphism, the only finite-dimensional simple associative algebras over the field '''C''' of [[complex number]]s. Prior to the invention of matrix algebras, [[William Rowan Hamilton|Hamilton]] in 1853 introduced a ring, whose elements he called [[biquaternions]]<ref>Lecture VII of Sir William Rowan Hamilton (1853) ''Lectures on Quaternions'', Hodges and Smith</ref> and modern authors would call tensors in {{nowrap|'''C''' ⊗<sub>'''R'''</sub> '''H'''}}, that was later shown to be isomorphic to M<sub>2</sub>('''C''').  One [[basis (linear algebra)|basis]] of M<sub>2</sub>('''C''') consists of the four matrix units (matrices with one 1 and all other entries 0); another basis is given by the [[identity matrix]] and the three [[Pauli matrices]].
* A matrix ring over a field is a [[Frobenius algebra]], with Frobenius form given by the trace of the product: {{nowrap|1=''σ''(''A'', ''B'') = tr(''AB'')}}.