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 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 operator 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 θ {displaystyle heta } is the angle of rotation. In the new coordinate system, the transformation would be written as where x' and y' are 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 x ′ = P x {displaystyle x'=Px} and y ′ = P y {displaystyle y'=Py} : Thus, the matrix in the original basis is given by T = P − 1 S P {displaystyle T=P^{-1}SP} . 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.