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Matrix consimilarity

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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.

Like ordinary similarity, consimilarity is an equivalence relation on the set of matrices, and it is reasonable to ask what properties it preserves.

The theory of ordinary similarity arises as a result of studying linear transformations referred to different bases. Consimilarity arises as a result of studying antilinear transformations referred to different bases.

A matrix is consimilar to itself, its complex conjugate, its transpose and its adjoint matrix. Every matrix is consimilar to a real matrix and to a Hermitian matrix. There is a standard form for the consimilarity class, analogous to the Jordan normal form.

References

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  • Hong, YooPyo; Horn, Roger A. (April 1988). "A canonical form for matrices under consimilarity". Linear Algebra and its Applications. 102: 143–168. doi:10.1016/0024-3795(88)90324-2. Zbl 0657.15008.
  • Horn, Roger A.; Johnson, Charles R. (1985). Matrix analysis. Cambridge: Cambridge University Press. ISBN 0-521-38632-2. Zbl 0576.15001. (sections 4.5 and 4.6 discuss consimilarity)