- #1
evalover1987
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Homework Statement
Homework Equations
The Attempt at a Solution
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arkajad said:I think that, excluding the zero matrix, for 2x2 matrices being rank 1 is equivalent to det(A)=0. That means tangent vectors are given by traceless matrices. That should be enough to solve your problem.
arkajad said:I was talking about tangent vectors - they should be traceless. See "http://www.math.ntnu.no/~hanche/notes/diffdet/diffdet-600dpi.ps" ".
arkajad said:I was talking about tangent vectors - they should be traceless. See "http://www.math.ntnu.no/~hanche/notes/diffdet/diffdet-600dpi.ps" ".
arkajad said:Another possible approach: A is of rank 1 if and only if there are unitary U,V such that
[tex]A=U\begin{pmatrix}1&0\\0&0\end{pmatrix}V[/tex]
Therefore a path in rank 1 matrices can be described as
[tex]A(t)=U(t)\begin{pmatrix}1&0\\0&0\end{pmatrix}V(t)[/tex]
Differentiating:
[tex]\dot{A}=\dot{U}U^{-1}A+AV^{-1}\dot{V}.[/tex]
Both [tex]\dot{U}U^{-1}[/tex] and [tex]V^{-1}\dot{V}[/tex]
are anti-hermitian. This gives you the general form of a tangent vector at A.
evalover1987 said:thanks for help, and I again apologize for my rudeness in the previous post
arkajad said:I was not very helpful. I tried to be, but I was too fast, and what I wrote is not yet a solution. But I think the solution is pretty close.
arkajad said:Take [tex]\begin{pmatrix}1&0\\0&-1\end{pmatrix}[/tex]. Exponentiate it to get:
[tex]S(t)=\begin{pmatrix}e^t&0\\0&e^{-t}\end{pmatrix}[/tex]
Calculate
[tex] S(t)\begin{pmatrix}0&0\\1&1\end{pmatrix}S(-t)=\begin{pmatrix}0&0\\e^{-2t}&1\end{pmatrix}[/tex]
P.S. Still confused ...
arkajad said:I have made several corrections in my example, but the last version may even work.
A tangent space is a mathematical concept used in differential geometry to describe the local behavior of curves and surfaces. It is a vector space that approximates the behavior of a curve or surface at a specific point.
A tangent space is defined as the set of all possible tangent vectors at a given point on a curve or surface. These tangent vectors represent the direction and rate of change of the curve or surface at that point.
The tangent space is important because it allows us to study the local behavior of a curve or surface. By understanding the tangent space, we can make predictions about the behavior of the curve or surface in the surrounding area.
The tangent space and the normal space are complementary concepts. The tangent space represents the direction of a curve or surface, while the normal space represents the direction perpendicular to the tangent space. Together, they provide a complete description of the local behavior of a curve or surface.
The tangent space has many practical applications in fields such as physics, engineering, and computer graphics. It is used to model and analyze the behavior of objects in motion, such as particles, fluids, and deformable structures. It is also used in computer graphics to generate realistic and smooth animations of 3D objects.