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A transformation A ↦ P −1 AP is called a similarity transformation or conjugation of the matrix A. In the general linear group , similarity is therefore the same as conjugacy , and similar matrices are also called conjugate ; however, in a given subgroup H of the general linear group, the notion of conjugacy may be more restrictive than ...
In linear algebra, similarity invariance is a property exhibited by a function whose value is unchanged under similarities of its domain. That is, f {\displaystyle f} is invariant under similarities if f ( A ) = f ( B − 1 A B ) {\displaystyle f(A)=f(B^{-1}AB)} where B − 1 A B {\displaystyle B^{-1}AB} is a matrix similar to A .
In linear algebra, two n-by-n matrices A and B are called consimilar if = ¯ for some invertible matrix , where ¯ denotes the elementwise complex conjugation.So for real matrices similar by some real matrix , consimilarity is the same as matrix similarity.
The trace is a map of Lie algebras : from the Lie algebra of linear operators on an n-dimensional space (n × n matrices with entries in ) to the Lie algebra K of scalars; as K is Abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement that the trace of a bracket vanishes: ([,]) =,.
Cosine similarity is the cosine of the angle between the vectors; that is, it is the dot product of the vectors divided by the product of their lengths. It follows that the cosine similarity does not depend on the magnitudes of the vectors, but only on their angle. The cosine similarity always belongs to the interval [,].
This is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector spaces and through matrices. Linear equations
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.
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.It has the determinant and the trace of the matrix among its coefficients.
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