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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.
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.
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).
The match/mismatch ratio of the matrix sets the target evolutionary distance. [8] [9] The +1/−3 DNA matrix used by BLASTN is best suited for finding matches between sequences that are 99% identical; a +1/−1 (or +4/−4) matrix is much more suited to sequences with about 70% similarity. Matrices for lower similarity sequences require longer ...
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.
The identity matrix commutes with all matrices. Jordan blocks commute with upper triangular matrices that have the same value along bands. If the product of two symmetric matrices is symmetric, then they must commute. That also means that every diagonal matrix commutes with all other diagonal matrices. [9] [10] Circulant matrices commute.
Let A be a square matrix. Then by Schur decomposition it is unitary similar to an upper-triangular matrix, say, B. If A is normal, so is B. But then B must be diagonal, for, as noted above, a normal upper-triangular matrix is diagonal. The spectral theorem permits the classification of normal matrices in terms of their spectra, for example:
Then J 1 and J 2 are similar and have the same spectrum, including algebraic multiplicities of the eigenvalues. The procedure outlined in the previous paragraph can be used to determine the structure of these matrices. Since the rank of a matrix is preserved by similarity transformation, there is a bijection between the Jordan blocks of J 1 and ...