搜搜做题作业网不是高手就别来浪费时间了～日,还是英文的.
来源：学生作业帮 编辑：搜搜做题作业网作业帮 分类：综合作业 时间：2024/07/22 22:48:02
不是高手就别来浪费时间了～日,还是英文的.
做的好的再加100分.
做的好的再加100分.
一万年已到!
For rotational matrixA,let x=(rcosa,rsina)
(a) Ax=A(rcosa,rsina)=(rcosθcosa-rsinθ sina,rsinθcosa+rcosθ sina)
=(rcos(a+θ),rsin(a+θ))
From the above equation,if A has real eigenvalue,θ=0 or 2π.
The eigenvalue is 1,eigen space is (rcosa,rsina) for all a.
(b)A has an eigenvalue i,
The characteristic equation det(tI-A)=t^2-2tcos(θ)+1=0
t=i ==>cos(θ)=0,i.e.,t^2+1=0,for eigenvalue=i,
A(x,y)=i(x,y)
(xcosθ-ysinθ,xsinθ+ycosθ)=(-y,x)=i(x,y)
eigenvector=(x,y)=(1 ,-i)
(c) determinant of A=(cosθ)^2+(sinθ)^2=1
The other eigenvalue=1/i=-i
A(x,y)=-i(x,y)
(-y,x)=-i(x,y)
eigenvector=(x,y)=(1,i)
再问： 给力，忍不住表扬你两句
再答： 咱们一起给力
再问： (-y,x)=i(x,y) eigenvector=(x,y)=(1 ,-i) 这结果怎么从上一步得出的？
再答： 设x=1 ==>y=-i, one of the eigenvector in eigen space
For rotational matrixA,let x=(rcosa,rsina)
(a) Ax=A(rcosa,rsina)=(rcosθcosa-rsinθ sina,rsinθcosa+rcosθ sina)
=(rcos(a+θ),rsin(a+θ))
From the above equation,if A has real eigenvalue,θ=0 or 2π.
The eigenvalue is 1,eigen space is (rcosa,rsina) for all a.
(b)A has an eigenvalue i,
The characteristic equation det(tI-A)=t^2-2tcos(θ)+1=0
t=i ==>cos(θ)=0,i.e.,t^2+1=0,for eigenvalue=i,
A(x,y)=i(x,y)
(xcosθ-ysinθ,xsinθ+ycosθ)=(-y,x)=i(x,y)
eigenvector=(x,y)=(1 ,-i)
(c) determinant of A=(cosθ)^2+(sinθ)^2=1
The other eigenvalue=1/i=-i
A(x,y)=-i(x,y)
(-y,x)=-i(x,y)
eigenvector=(x,y)=(1,i)
再问： 给力，忍不住表扬你两句
再答： 咱们一起给力
再问： (-y,x)=i(x,y) eigenvector=(x,y)=(1 ,-i) 这结果怎么从上一步得出的？
再答： 设x=1 ==>y=-i, one of the eigenvector in eigen space