w***z
发帖数: 771 | 1 a rotation matrix A : | cos(t), -sin(t) |, t is the rotation angle
| sin(t), cos(t) |
has two complex eigenvalue r = cos(t) +- sin(t)*i
and two complex eigenvector e = [1, i], [1, -i]
if t = 90 degree,
we have r = i, -i;
and e = [1,i], [1,-i]
My question is how we understand e keeps the same direction for a 90 degree
rotation matrix, or any rotation matrix? what's the intuition here? Looks
like we need four-dimension to imagine it. any idea to imagine it? | n*****n
发帖数: 5277 | 2 geometry meaning of eigenvector in two dimensional space is not as
straightforward as what in 3 dimensional space. I guess the changing part
has been included in eigenvalue. | L********r
发帖数: 758 | 3 It's the same reason as why Fourier transform formula is always the same for
different LTI systems. You can think of the eigen vector matrix as your
portal from one universe to the other. The first universe describes a class
of operators in a complicated way, while the second describes the same class
of operators in a simple way. The operators in this class share the same
portal.
degree
【在 w***z 的大作中提到】
: a rotation matrix A : | cos(t), -sin(t) |, t is the rotation angle
: | sin(t), cos(t) |
: has two complex eigenvalue r = cos(t) +- sin(t)*i
: and two complex eigenvector e = [1, i], [1, -i]
: if t = 90 degree,
: we have r = i, -i;
: and e = [1,i], [1,-i]
: My question is how we understand e keeps the same direction for a 90 degree
: rotation matrix, or any rotation matrix? what's the intuition here? Looks
: like we need four-dimension to imagine it. any idea to imagine it?
| w***z
发帖数: 771 | 4 a rotation matrix A : | cos(t), -sin(t) |, t is the rotation angle
| sin(t), cos(t) |
has two complex eigenvalue r = cos(t) +- sin(t)*i
and two complex eigenvector e = [1, i], [1, -i]
if t = 90 degree,
we have r = i, -i;
and e = [1,i], [1,-i]
My question is how we understand e keeps the same direction for a 90 degree
rotation matrix, or any rotation matrix? what's the intuition here? Looks
like we need four-dimension to imagine it. any idea to imagine it? | n*****n
发帖数: 5277 | 5 geometry meaning of eigenvector in two dimensional space is not as
straightforward as what in 3 dimensional space. I guess the changing part
has been included in eigenvalue. | L********r
发帖数: 758 | 6 It's the same reason as why Fourier transform formula is always the same for
different LTI systems. You can think of the eigen vector matrix as your
portal from one universe to the other. The first universe describes a class
of operators in a complicated way, while the second describes the same class
of operators in a simple way. The operators in this class share the same
portal.
degree
【在 w***z 的大作中提到】
: a rotation matrix A : | cos(t), -sin(t) |, t is the rotation angle
: | sin(t), cos(t) |
: has two complex eigenvalue r = cos(t) +- sin(t)*i
: and two complex eigenvector e = [1, i], [1, -i]
: if t = 90 degree,
: we have r = i, -i;
: and e = [1,i], [1,-i]
: My question is how we understand e keeps the same direction for a 90 degree
: rotation matrix, or any rotation matrix? what's the intuition here? Looks
: like we need four-dimension to imagine it. any idea to imagine it?
|
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