row) to , so that the remainder of the matrix is a rotation matrix of dimension one less, embedded so as to leave fixed. Skew parameters via Cayley's formula When an rotation matrix , does not include a −1 eigenvalue, thus none of the planar rotations which it comprises are 180° rotations, then is an invertible matrix. Most rotation matrices fit this description, and for them it can be shown that is a skew-symmetric matrix, . Thus ; and since the diagonal is necessarily zero, and since the upper triangle determines the lower one, contains independent numbers. Conveniently, is invertible whenever is skew-symmetric; thus we can recover the original matrix using the Cayley transform, which maps any skew-symmetric matrix to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although in practical applications we can hardly afford to ignore 180° rotations, the Cayley transform is still a potentially useful tool, giving a parameterization of most rotation matrices without trigonometric functions. In three dimensions, for example, we have If we condense the skew entries into a vector, , then we produce a 90° rotation around the -axis for (1, 0, 0), around the -axis for (0, 1, 0), and around the -axis for (0, 0, 1). The 180° rotations are just out of reach; for, in the limit as , does approach a 180° rotation around the axis, and similarly for other directions. Decomposition into shears For the 2D case, a rotation matrix can be decomposed into three shear matrices (): This is useful, for instance, in computer graphics, since shears can be implemented with fewer multiplication instructions than rotating a bitmap directly. On modern computers, this may not matter, but it can be relevant for very old or low-end microprocessors. A rotation can also be written as two shears and scaling (): Group theory Below follow some basic facts about the role of the collection of all rotation matrices of a fixed dimension (here mostly 3) in mathematics and particularly in physics where rotational symmetry is a requirement of every truly fundamental law (due to the assumption of isotropy of space), and where the same symmetry, when present, is a simplifying property of many problems of less fundamental nature. Examples abound in classical mechanics and quantum mechanics. Knowledge of the part of the solutions pertaining to this symmetry applies (with