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Elementary divisors, which form a complete set of invariants for similarity of matrices over a principal ideal domain; Because of this, for a given matrix A, one is interested in finding a simple "normal form" B which is similar to A—the study of A then reduces to the study of the simpler matrix B.
Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases.
So for real matrices similar by some real matrix , consimilarity is the same as matrix similarity. Like ordinary similarity, consimilarity is an equivalence relation on the set of n × n {\displaystyle n\times n} matrices, and it is reasonable to ask what properties it preserves.
Similarity (geometry), the property of sharing the same shape; Matrix similarity, a relation between matrices; Similarity measure, a function that quantifies the similarity of two objects Cosine similarity, which uses the angle between vectors; String metric, also called string similarity; Semantic similarity, in computational linguistics
In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if = for some invertible n-by-n matrix P and some invertible m-by-m matrix Q.Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.
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In data analysis, the self-similarity matrix is a graphical representation of similar sequences in a data series. Similarity can be explained by different measures, like spatial distance ( distance matrix ), correlation , or comparison of local histograms or spectral properties (e.g. IXEGRAM [ 1 ] ).