an angle , or a rotation of by the same angle but in the opposite direction (i.e. clockwise). Alibi and alias transformations are also known as active and passive transformations, respectively. Pre-multiplication or post-multiplication The same point can be represented either by a column vector or a row vector . Rotation matrices can either pre-multiply column vectors (), or post-multiply row vectors (). However, produces a rotation in the opposite direction with respect to . Throughout this article, rotations produced on column vectors are described by means of a pre-multiplication. To obtain exactly the same rotation (i.e. the same final coordinates of point ), the equivalent row vector must be post-multiplied by the transpose of (i.e. ). Right- or left-handed coordinates The matrix and the vector can be represented with respect to a right-handed or left-handed coordinate system. Throughout the article, we assumed a right-handed orientation, unless otherwise specified. Vectors or forms The vector space has a dual space of linear forms, and the matrix can act on either vectors or forms. Decompositions Independent planes Consider the rotation matrix If acts in a certain direction, , purely as a scaling by a factor , then we have so that Thus is a root of the characteristic polynomial for , Two features are noteworthy. First, one of the roots (or eigenvalues) is 1, which tells us that some direction is unaffected by the matrix. For rotations in three dimensions, this is the axis of the rotation (a concept that has no meaning in any other dimension). Second, the other two roots are a pair of complex conjugates, whose product is 1 (the constant term of the quadratic), and whose sum is (the negated linear term). This factorization is of interest for rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthermore, a similar factorization holds for any rotation matrix. If the dimension, , is odd, there will be a "dangling" eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the two special cases noted). We are guaranteed that the characteristic polynomial will have degree and thus eigenvalues. And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized.