Symbolic Math Toolbox 5.3

Plane Rotations

Create a symbolic variable named t.

t = sym('t')

t =

t

Create a 2-by-2 matrix representing a plane rotation through an angle t.

G = [ cos(t) sin(t); -sin(t) cos(t)]

G =

[  cos(t), sin(t)]
[ -sin(t), cos(t)]

Compute the matrix product of G with itself.

G*G

ans =

[ cos(t)^2 - sin(t)^2,     2*cos(t)*sin(t)]
[  (-2)*cos(t)*sin(t), cos(t)^2 - sin(t)^2]

This should represent a rotation through an angle of 2*t. Simplification using trigonometric identities is necessary.

ans = simple(ans)

ans =

[  cos(2*t), sin(2*t)]
[ -sin(2*t), cos(2*t)]

G is an orthogonal matrix; its transpose is its inverse.

G.'*G

ans = simple(ans)

ans =

[ cos(t)^2 + sin(t)^2,                   0]
[                   0, cos(t)^2 + sin(t)^2]


ans =

[ 1, 0]
[ 0, 1]

What are the eigenvalues of G?

e = eig(G)

e =

 cos(t) - i*sin(t)
 cos(t) + i*sin(t)

Repeatedly apply the simplification rules.

e, for k = 1:4, e = simple(e), end

e =

 cos(t) - i*sin(t)
 cos(t) + i*sin(t)


e =

 1/exp(i*t)
   exp(i*t)


e =

 1/exp(i*t)
   exp(i*t)


e =

 1/exp(i*t)
   exp(i*t)


e =

 1/exp(i*t)
   exp(i*t)