no axis plane. The case of is called an isoclinic rotation, having eigenvalues repeated twice, so every vector is rotated through an angle . The trace of a rotation matrix is equal to the sum of its eigenvalues. For , a rotation by angle has trace . For , a rotation around any axis by angle has trace . For , and the trace is , which becomes for an isoclinic rotation. Examples The rotation matrix corresponds to a 90° planar rotation clockwise about the origin. The transpose of the matrix is its inverse, but since its determinant is −1, this is not a proper rotation matrix; it is a reflection across the line . The rotation matrix corresponds to a −30° rotation around the -axis in three-dimensional space. The rotation matrix corresponds to a rotation of approximately −74° around the axis in three-dimensional space. The permutation matrix is a rotation matrix, as is the matrix of any even permutation, and rotates through 120° about the axis . The matrix has determinant +1, but is not orthogonal (its transpose is not its inverse), so it is not a rotation matrix. The matrix is not square, and so cannot be a rotation matrix; yet yields a identity matrix (the columns are orthonormal). The matrix describes an isoclinic rotation in four dimensions, a rotation through equal angles (180°) through two orthogonal planes. The rotation matrix rotates vectors in the plane of the first two coordinate axes 90°, rotates vectors in the plane of the next two axes 180°, and leaves the last coordinate axis unmoved. Geometry In Euclidean geometry, a rotation is an example of an isometry, a transformation that moves points without changing the distances between them. Rotations are distinguished from other isometries by two additional properties: they leave (at least) one point fixed, and they leave "handedness" unchanged. In contrast, a translation moves every point, a reflection exchanges left- and right-handed ordering, a glide reflection does both, and an improper rotation combines a change in handedness with a normal rotation. If a fixed point is taken as the origin of a Cartesian coordinate system, then every point can be given coordinates as a displacement from the origin. Thus one may work with the vector space of displacements instead of the points themselves. Now suppose are the coordinates of the vector from the origin to point . Choose an orthonormal basis