Contents
Can a real matrix have complex eigenvectors?
In general, a real matrix can have a complex number eigenvalue. In fact, the part (b) gives an example of such a matrix.
Is the rotation matrix unique?
Are rotation matrices unique? Yes they are, as this answer that Francesco quoted explains well. If they were not unique, then Qv = Rv and thus (Q-R)*v = 0 would be true for any vector. The latter is only true for the null matrix, however.
Can a matrix with repeated eigenvalues be diagonalizable?
A matrix with repeated eigenvalues can be diagonalized. Just think of the identity matrix. All of its eigenvalues are equal to one, yet there exists a basis (any basis) in which it is expressed as a diagonal matrix.
Is the Convention of a rotation matrix the same?
The convention of these rotation matrices follow the same positive direction of rotation as the cross product in a right handed coordinate system.
Why do I See Your C in my rotation matrix?
If you read your rotation matrix from left to right, what appears is the X-Z coordinate system instead of the Z-X coordinate system, also along Y, but now we see that Y is pointing away from you now. This changes the direction of rotation. Hence, you see R c appearing instead of R c c.
How are rotation matrices used in computer graphics?
Since matrix multiplication has no effect on the zero vector(the coordinates of the origin), rotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics.
How is the position of a point represented in a rotation matrix?
rotates points in the xy-plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point.