Reduction of a matrix depending on parameters to a diagonal form by addition operations

Vaserstein, L. N.

doi:10.1090/S0002-9939-1988-0947649-X

Reduction of a matrix depending on parameters to a diagonal form by addition operations
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Proc. Amer. Math. Soc. 103 (1988), 741-746 Request permission

Abstract:

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.

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