Hi all, I am trying to construct a continuous matrix-valued function h=h(f,g) ; f,g in C^1[a,b],all defined in an open interval (a,b) on the Real line with some given properties, or at least I want to show that the function exists . I have/know that for a,b in [a,b], both Det(h(a))>0 and Det(h(b))>0 . Then I want, for all t in (a,b), that the determinant of :(adsbygoogle = window.adsbygoogle || []).push({});

[tex] A=

\begin{pmatrix}

f(t) & g(t) \\

f '(t)& g'(t) \\

\end{pmatrix}

[/tex]

will be positive . I am trying to use that GL(n,R) has two connected components; GL^+ and

Gl^- ;matrices with positive and negative determinant respectively (e.g., use the fact that

Gaussian elimination can take any A in GL^+ into the Id. ).

Then I think can argue that since f,g ;f',g' are continuous by assumption, the map h(f,g) is

itself also continuous , so that h[a,b] -- remember here that [a,b] is an interval in the real line--is connected, and so h[a,b] must then lie in GL^+(n,R), for all t in [a,b], i.e., Det(h[a,b])>0.

Is this right?

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# Paths within GL^+(n,R)

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