Sending a matrix to the endomorphism \(v\mapsto Av\) is an isomorphism of \(K\)-algebras \(M_n(K)\to \operatorname{End}_{K}(K^n)\), and the identity map of \(K^n\) is semilinear along it, since it carries the matrix action to the endomorphism action. A nonzero vector space over a division ring is a simple module over its own endomorphism ring, and simplicity transports along that semilinear bijection. Schumacher and Werner call \(K^n\) the defining representation of \(M_n(K)\) at [ 237 , lines 2112–2114 ] .

A simple Artinian ring is semisimple, so a simple module over it is isomorphic to a simple left ideal: \(M\cong I\) and \(N\cong J\) with \(I,J\subseteq R\) simple left ideals. The regular module of a simple Artinian ring is isotypic, so \(I\cong J\), and composing gives \(M\cong N\). This is the general form of the property of \(M_{d}(\mathbb {C})\) that all its irreducible representations are unitarily equivalent to the defining one, quoted at [ 237 , lines 2112–2114 ] .

Give \(\mathbb {C}^q\) the \(M_{p}(\mathbb {C})\)-action \(A\cdot v=\Phi (A)v\); it is compatible with the complex scalars because \(\Phi \) is complex linear. Let \(W\subseteq \mathbb {C}^q\) be an \(M_{p}(\mathbb {C})\)-submodule, choose an \(M_{p}(\mathbb {C})\)-submodule complement \(W^{\prime }\) with \(\mathbb {C}^q=W\oplus W^{\prime }\), and let \(P\) be the projection onto \(W\) along \(W^{\prime }\), so that \(W=\operatorname{ran}P\). Since \(P\) is \(M_{p}(\mathbb {C})\)-linear, \(P\Phi (A)=\Phi (A)P\) for every \(A\in M_{p}(\mathbb {C})\), so \(P\) lies in (1) and hence \(P=c\, \mathbb {1}_q\) for some \(c\in \mathbb {C}\). If \(c=0\) then \(P=0\) and \(W=0\); if \(c\neq 0\) then \(v=P(c^{-1}v)\) for every \(v\in \mathbb {C}^q\) and \(W=\mathbb {C}^q\). So \(\mathbb {C}^q\) is a simple \(M_{p}(\mathbb {C})\)-module. The ring \(M_{p}(\mathbb {C})\) is simple Artinian and \(\mathbb {C}^p\) is a simple \(M_{p}(\mathbb {C})\)-module by Lemma H.1, so Lemma H.2 gives an \(M_{p}(\mathbb {C})\)-linear isomorphism \(\mathbb {C}^q\cong \mathbb {C}^p\). Reading it as an isomorphism of complex vector spaces and comparing dimensions gives \(q=p\). This is the single-summand, multiplicity-one case of the representation argument at [ 237 , lines 2101–2116 ] .

The kernel of \(\Phi \) is a two-sided ideal of the simple algebra \(M_{p}(\mathbb {C})\), and it is not all of \(M_{p}(\mathbb {C})\) because \(\Phi (\mathbb {1}_p)=\mathbb {1}_q\neq 0\); hence \(\Phi \) is injective. By Theorem H.3, \(p=q\), so \(M_{p}(\mathbb {C})\) and \(M_{q}(\mathbb {C})\) have the same finite complex dimension, and an injective complex-linear map between complex vector spaces of equal finite dimension is onto.

The involution plays no part in the argument, so Theorem H.4 applies to \(\Phi \) read as a homomorphism of complex algebras.

If every \(T\) commuting with the whole image of \(\Phi \) were scalar, then (1) would equal \(\mathbb {C}\mathbb {1}_q\) and Theorem H.4 would make \(\Phi \) onto.