Matrix similarity

In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that Similar matrices represent the same linear map under two (possibly) different bases, with P being the change of basis matrix. A transformation A ↦ P−1AP is called a similarity transformation or conjugation of the matrix A. In the general linear group, similarity is therefore the same as conjugacy, and similar matrices are also called conjugate; however, in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than similarity, since it requires that P be chosen to lie in H. When defining a linear transformation, it can be the case that a change of basis can result in a simpler form of the same transformation. For example, the matrix representing a rotation in R3 when the axis of rotation is not aligned with the coordinate axis can be complicated to compute. If the axis of rotation were aligned with the positive z-axis, then it would simply be where is the angle of rotation. In the new coordinate system, the transformation would be written as where x' and y' are respectively the original and transformed vectors in a new basis containing a vector parallel to the axis of rotation. In the original basis, the transform would be written as where vectors x and y and the unknown transform matrix T are in the original basis. To write T in terms of the simpler matrix, we use the change-of-basis matrix P that transforms x and y as and : Thus, the matrix in the original basis, , is given by . The transform in the original basis is found to be the product of three easy-to-derive matrices. In effect, the similarity transform operates in three steps: change to a new basis (P), perform the simple transformation (S), and change back to the old basis (P−1). Similarity is an equivalence relation on the space of square matrices.