# Matrix similarity

## Equivalence under a change of basis (linear algebra) / From Wikipedia, the free encyclopedia

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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.[1][2]

A transformation *A* ↦ *P*^{−1}*AP* 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.