Matrix consimilarity
In linear algebra, two n-by-n matrices A and B are called consimilar if
- [math]\displaystyle{ A = S B \bar{S}^{-1} \, }[/math]
for some invertible [math]\displaystyle{ n \times n }[/math] matrix [math]\displaystyle{ S }[/math], where [math]\displaystyle{ \bar{S} }[/math] denotes the elementwise complex conjugation. So for real matrices similar by some real matrix [math]\displaystyle{ S }[/math], consimilarity is the same as matrix similarity.
Like ordinary similarity, consimilarity is an equivalence relation on the set of [math]\displaystyle{ n \times n }[/math] 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
- 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.
- Horn, Roger A.; Johnson, Charles R. (1985). Matrix analysis. Cambridge: Cambridge University Press. ISBN 0-521-38632-2. (sections 4.5 and 4.6 discuss consimilarity)
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