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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 so because the rational canonical form over K is also the rational canonical form over L. This means ...
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.
Circular matrix or Coninvolutory matrix: A matrix whose inverse is equal to its entrywise complex conjugate: A −1 = A. Compare with unitary matrices. Congruent matrix: Two matrices A and B are congruent if there exists an invertible matrix P such that P T A P = B. Compare with similar matrices. EP matrix or Range-Hermitian matrix
Because the null space of a matrix is the orthogonal complement of the row space, two matrices are row equivalent if and only if they have the same null space. The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form.
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.
Informally speaking, every matrix satisfies its own characteristic equation. This statement is equivalent to saying that the minimal polynomial of divides the characteristic polynomial of . Two similar matrices have the same characteristic polynomial. The converse however is not true in general: two matrices with the same characteristic ...
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.
These matrices are related as follows. The following statements are equivalent: A is similar over F to (), i.e. A can be conjugated to its companion matrix by matrices in GL n (F); the characteristic polynomial () coincides with the minimal polynomial of A, i.e. the minimal polynomial has degree n;