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Here’s a quickie: What are the eigenvalues of a 2D rotation matrix?
Here’s a problem: For a bunch of rotations performed one after another on a 3D object, find an equivalent single rotation which would give the same result.
Here’s a solution: First of all, multiply all the rotation matrices (obviously!) such that . A single rotation is characterized by an axis and an angle of rotation. Let’s get the angle first. Find the eigenvalues of . Two of them will be of the form: . This θ is the required angle. Let’s get the axis now. So what would be the third eigenvalue? Remember which means the third one is 1! Find the eigenvector corresponding to this eigenvalue which is the required axis!
Here’s an example: Let’s do the problem mentioned in Gravitation.
Relevant eigenvalues of are which give us the angle: . The eigenvector corresponding to 1 is so the axis is one of the diagonals! This solution agrees with the one given in the book.