Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In linear algebra, two n-by-n matrices A and B are called similar if ! B = P^{-1} A P for some invertible n-by-n matrix P. Similar matrices represent the same linear transformation under two different bases, with P being the change of basis matrix. The matrix P is sometimes called a similarity transformation. In the context of matrix groups, similarity is sometimes referred to as conjugacy, with similar matrices being conjugate. Similarity of matrices does not depend on the base field: if L is a field containing K as a subfield, and A and B are two matrices over K, then A and B are similar as matrices over K if and only if they are similar as matrices over L. This is quite useful: one may safely enlarge the field K, for instance to get an algebraically closed field; Jordan forms can then be computed over the large field and can be used to determine whether the given matrices are similar over the small field. This approach can be used, for example, to show that every matrix is similar to its transpose.

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