Answer to Question #1452 in Linear Algebra for AA

Question

A = 4 2
-5 -2

Find an invertible 2x2 matrix Q such that Q^-1AQ is equal to a rotation followed by scaling by a positive real number, and write down both the rotation matrix and the scaling matrix. What is the angle of rotation of the matrix?

Answer 1

Rotation M = Scaling M
cos(theta) -sin(theta) = s_x 0
sin(theta) cos(theta) = 0 s_y
Thus
Q= 1 0
0 1
Scalling matrix is the same as well as rotation matrix. As cos(theta) = 1, rotation angle is equal to zero.