How do rotations change in four dimensions?
Compare a fixed axis in three dimensions with planes of rotation in four. The familiar axis description does not carry over unchanged.
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Why shift a matrix before looking for an eigenvector? →A zero determinant signals collapse into a lower dimension. Applying that test to the shifted matrix explains how a nonzero vector can lie in its null space.Which outputs disappear when a transformation collapses space? →Some target vectors fall outside the image after a collapse. In the eigenvalue example it is A minus lambda times the identity that becomes singular; this does not make every system unsolvable.
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what you have found is the axis of rotationtwo separate 2D rotations, happening perpendicular to and in sync with each other