Reduction of a Matrix Depending on Parameters to a Diagonal Form by Addition Operations

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It is shown that any $n$ by $n$ matrix with determinant 1 whose entries are real or complex continuous functions on a finite dimensional normal topological space can be reduced to a diagonal form by addition operations if and only if the corresponding homotopy class is trivial, provided that $n \ne 2$ for real-valued functions; moreover, if this is the case, the number of operations can be bounded by a constant depending only on $n$ and the dimension of the space. For real functions and $n = 2$, we describe all spaces such that every invertible matrix with trivial homotopy class can be reduced to a diagonal form by addition operations as well as all spaces such that the number of operations is bounded.