Let $X$ be a compact metric space which is locally absolutely retract and let $phi: C(X)to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact metric space with ${rm dim}Yle 2.$ It is proved that there exists a sequence of $n$ continuous maps $alfa_{i,m}: Yto X$ ($i=1,2,...,n$) and a sequence of sets of mutually orthogonal rank one projections ${p_{1, m}, p_{2,m},...,p_{n,m}}subset C(Y, M_n)$ such that $$ lim_{mtoinfty} sum_{i=1}^n f(alfa_{i,m})p_{i,m}=phi(f) for all fin C(X). $$ This is closely related to the Kadison diagonal matrix question. It is also shown that this approximate diagonalization could not hold in general when ${rm dim}Yge 3.$
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