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This means that one may use Jordan forms that only exist over a larger field to determine whether the given matrices are similar. In the definition of similarity, if the matrix P can be chosen to be a permutation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and B are unitarily equivalent.
Similarity Matrix of Proteins (SIMAP) is a database of protein similarities created using volunteer computing. [ 1 ] [ 2 ] It is freely accessible for scientific purposes. SIMAP uses the FASTA algorithm to precalculate protein similarity, while another application uses hidden Markov models to search for protein domains .
E.g., BLOSUM62 is the matrix built using sequences with less than 62% similarity (sequences with ≥ 62% identity were clustered together). Note: BLOSUM 62 is the default matrix for protein BLAST. Experimentation has shown that the BLOSUM-62 matrix is among the best for detecting most weak protein similarities. [1]
Not every square matrix is similar to a companion matrix, but every square matrix is similar to a block diagonal matrix made of companion matrices. If we also demand that the polynomial of each diagonal block divides the next one, they are uniquely determined by A, and this gives the rational canonical form of A.
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.
This PAM1 matrix estimates what rate of substitution would be expected if 1% of the amino acids had changed. The PAM1 matrix is used as the basis for calculating other matrices by assuming that repeated mutations would follow the same pattern as those in the PAM1 matrix, and multiple substitutions can occur at the same site.
Two similar matrices represent the same linear map, but with respect to a different basis; unitary equivalence corresponds to a change from an orthonormal basis to another orthonormal basis. If A and B are unitarily equivalent, then tr AA * = tr BB *, where tr denotes the trace (in other words, the Frobenius norm is a unitary invariant).
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