- #1
- 7,006
- 10,453
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 :
[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?
[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?