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Concept# Matrix similarity

Summary

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
B = P^{-1} A P .
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.
Motivating example
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 representin

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