Rotation matrices

1

Constructing rotation matrices Eigenvectors and eigenvalues

0x

y

𝑣

𝑣 ′

𝜙𝜃

𝑣 ′=�̂� (𝜃 ) �⃑�

𝜆±=𝑒±𝑖 𝜃

[𝑣𝑥±

𝑣 𝑦± ]=[ 1

∓𝑖 ]

�̂�𝑁 (𝜃 ) �⃑�→𝑤+¿𝑒+ 𝑖 𝑁𝜃 [ 1

−𝑖 ]+𝑤−𝑒−𝑖 𝑁 𝜃[ 1

+𝑖 ]¿

2

Rotating a vector

𝑣→[𝑣 c os𝜙𝑣 sin 𝜙 ]=[𝑣𝑥

𝑣𝑦]

𝑣 ′→[𝑣 c os (𝜙+𝜃 )𝑣 sin (𝜙+𝜃 ) ]

0x

y

𝑣

𝑣 ′

𝜙𝜃

Leng

th v

(cos𝜙 cos𝜃− sin𝜙 sin𝜃 )(sin 𝜙cos𝜃+cos𝜙 sin 𝜃 )

𝑣 ′→[𝑣 cos𝜙cos𝜃−𝑣 sin 𝜙 sin 𝜃𝑣 sin 𝜙cos𝜃+𝑣 cos𝜙 sin𝜃 ]𝑣 𝑥

𝑣 𝑦𝑣 𝑥

𝑣 𝑦

𝑣 ′→[𝑣 𝑥cos𝜃−𝑣𝑦 sin 𝜃𝑣𝑥sin 𝜃+𝑣 𝑦cos𝜃 ]

𝑣 ′→[cos𝜃 −sin 𝜃sin 𝜃 cos𝜃 ] [𝑣 𝑥

𝑣 𝑦][𝑣𝑥

′

𝑣 𝑦′ ]=¿

𝑣 ′=�̂� (𝜃 ) �⃑�

3

Repeatedly rotating a vector

0x

y

𝑣

𝜙𝜃

[𝑣𝑥′

𝑣 𝑦′ ]=[cos𝜃 − sin𝜃

sin𝜃 cos𝜃 ][cos𝜃 −sin 𝜃sin 𝜃 cos𝜃 ] [cos𝜃 −sin 𝜃

sin 𝜃 cos𝜃 ] [cos𝜃 − sin𝜃sin𝜃 cos𝜃 ][𝑣𝑥

𝑣 𝑦 ]𝑣 ′=�̂� (𝜃 ) �⃑�𝑣 ′=�̂�2 (𝜃 ) �⃑�

𝑣 ′=�̂�3 (𝜃 ) �⃑�

𝑣 ′=�̂�4 (𝜃 )𝑣

𝜃𝜃

𝜃

[𝑣𝑥 (𝑡+Δ𝑡 )𝑣 𝑦 (𝑡+Δ 𝑡 )]=[cos𝜃 − sin 𝜃

sin 𝜃 cos𝜃 ][𝑣𝑥 (𝑡 )𝑣𝑦 (𝑡 )]

[𝑣𝑥 (𝑡+Δ𝑡 )𝑣 𝑦 (𝑡+Δ 𝑡 )]=[0.87 −0.5

0.5 0.87 ][𝑣𝑥 (𝑡 )𝑣𝑦 (𝑡 )]

[𝑣𝑥 (𝑡+Δ𝑡 )𝑣 𝑦 (𝑡+Δ 𝑡 )]=[𝑎 𝑓 𝑔

h 𝑏2 𝑗 ] [𝑣𝑥 (𝑡 )𝑣𝑦 (𝑡 )]

STOP How can we recognize a rotation matrix?

Rotation matrices

4

Constructing rotation matrices Eigenvectors and eigenvalues

0x

y

𝑣

𝑣 ′

𝜙𝜃

𝑣 ′=�̂� (𝜃 ) �⃑�

𝜆±=𝑒±𝑖 𝜃

�̂�𝑁 (𝜃 ) �⃑�→𝑤+¿𝑒+ 𝑖 𝑁𝜃 [ 1

−𝑖 ]+𝑤−𝑒−𝑖 𝑁 𝜃[ 1

+𝑖 ]¿

[𝑣𝑥±

𝑣 𝑦± ]=[ 1

∓𝑖 ]

5

Complex eigenvalues and eigenvectors

[𝑣𝑥 (𝑡+Δ𝑡 )𝑣 𝑦 (𝑡+Δ 𝑡 )]=[cos𝜃 − sin 𝜃

sin 𝜃 cos𝜃 ][𝑣𝑥 (𝑡 )𝑣𝑦 (𝑡 )]

(cos𝜃− 𝜆 ) (cos𝜃− 𝜆 )+sin2𝜃=0

𝜆2−2𝜆cos𝜃+1=0𝜆±=cos𝜃± 𝑖sin 𝜃STOP Please confirm

these last 2 lines

𝜆±=𝑒±𝑖 𝜃

Complex eigenvalues

( cos𝜃− 𝜆± )𝑣𝑥−sin 𝜃𝑣 𝑦=0

[𝑣𝑥±

𝑣 𝑦± ]=[ 1

∓𝑖 ] Complex eigenvectors

�⃑�→𝑤+¿[ 1

−𝑖 ]+𝑤−[ 1+𝑖 ]¿

6

Complex eigenvalues and eigenvectors

[𝑣𝑥 (𝑡+Δ𝑡 )𝑣 𝑦 (𝑡+Δ 𝑡 )]=[cos𝜃 − sin 𝜃

sin 𝜃 cos𝜃 ][𝑣𝑥 (𝑡 )𝑣𝑦 (𝑡 )]

𝜆±=𝑒±𝑖 𝜃

Complex eigenvalues

[𝑣𝑥±

𝑣 𝑦± ]=[ 1

∓𝑖 ] Complex eigenvectors

�⃑�→𝑤+¿[ 1

−𝑖 ]+𝑤−[ 1+𝑖 ]¿�̂� (𝜃 )

[cos𝜃 −sin 𝜃sin 𝜃 cos𝜃 ]

[cos𝜃 −sin 𝜃sin 𝜃 cos𝜃 ]

𝜆+¿¿𝜆+¿¿

𝜆+¿¿

[cos𝜃 −sin 𝜃sin 𝜃 cos𝜃 ]

𝜆−𝜆−𝜆−

�̂�𝑁 (𝜃 ) �⃑�→𝑤+¿𝜆+¿𝑁[ 1

− 𝑖]+𝑤−𝜆−𝑁 [ 1

+ 𝑖] ¿¿𝑒+𝑖 𝑁𝜃 𝑒−𝑖 𝑁 𝜃

7

Complex eigenvalues and eigenvectors

[𝑣𝑥 (𝑡+Δ𝑡 )𝑣 𝑦 (𝑡+Δ 𝑡 )]=[cos𝜃 − sin 𝜃

sin 𝜃 cos𝜃 ][𝑣𝑥 (𝑡 )𝑣𝑦 (𝑡 )]

�̂�𝑁 (𝜃 ) �⃑�→𝑤+¿𝑒+ 𝑖 𝑁 𝜃 [ 1

−𝑖 ]+𝑤−𝑒−𝑖 𝑁 𝜃[ 1

+𝑖 ]¿Consider an example with w+ = w- = w/2

�̂�𝑁 (𝜃 ) �⃑�→𝑤 [ 𝑒+𝑖 𝑁 𝜃+𝑒−𝑖 𝑁 𝜃

2

−𝑖 𝑖(𝑒+ 𝑖𝑁 𝜃−𝑒−𝑖 𝑁 𝜃

2𝑖 )]=𝑤[ cos𝑁𝜃sin𝑁𝜃 ] 0

x

y

�⃑�

(cos 𝑁𝜃+ 𝑖sin 𝑁𝜃 ) (cos 𝑁𝜃−𝑖 sin𝑁

𝜃 )

(cos 𝑁𝜃+ 𝑖sin 𝑁𝜃 )

𝑤2

𝑤2

(cos𝑁𝜃−

𝑖 sin𝑁 𝜃 )

1. In this example, the initial vector points directly to the right. How should the coefficients w+ and w- be changed to represent an initial vector pointing at an arbitrary initial angle relative to the x axis?

2. Can you plot the eigenvectors (1, ±i) on the xy plane? (No). Why not?