Unitary equivalence to a complex symmetric matrix: An algorithm

@article{Tener2008UnitaryET,
  title={Unitary equivalence to a complex symmetric matrix: An algorithm},
  author={James E. Tener},
  journal={Journal of Mathematical Analysis and Applications},
  year={2008},
  volume={341},
  pages={640-648}
}
  • James E. Tener
  • Published 1 May 2008
  • Mathematics
  • Journal of Mathematical Analysis and Applications
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References

SHOWING 1-10 OF 18 REFERENCES
Fast Diagonalization of Large and Dense Complex Symmetric Matrices, with Applications to Quantum Reaction Dynamics
TLDR
A new fast and efficient algorithm for computing the eigenvalues and eigenvectors of large-size nondefective complex symmetric matrices is presented, similiar to the QR (QL) algorithm for complex Hermitian matrices, but it uses complex orthogonal (not unitary) transformations.
THE CHARACTERISTIC FUNCTION OF A COMPLEX SYMMETRIC CONTRACTION
It is shown that a contraction on a Hilbert space is complex sym- metric if and only if the values of its characteristic function are all symmetric with respect to a fixed conjugation. Applications
Some new classes of complex symmetric operators
We say that an operator T E B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C: ℌ→ℌ so that T = CT * C. We prove that binormal operators, operators that are
Complex Symmetric Operators and Applications II
A bounded linear operator T on a complex Hilbert space H is called complex symmetric if T = CT*C, where C is a conjugation (an isometric, antilinear involution of H). We prove that T = CJ\T|, where J
Approximate antilinear eigenvalue problems and related inequalities
If T is a complex symmetric operator on a separable complex Hilbert space H, then the spectrum σ(|T|) of √T*T can be characterized in terms of a certain approximate antilinear eigenvalue problem.
Conjugation and Clark Operators
We discuss the application of antilinear symmetries (conjugation operators) to problems connected to the compressed shift on the spaces H φH where φ denotes a nonconstant inner function. For example,
Norm estimates of complex symmetric operators applied to quantum systems
This paper communicates recent results in the theory of complex symmetric operators and shows, through two non-trivial examples, their potential usefulness in the study of Schrodinger operators. In
Truncated Toeplitz Operators on Finite Dimensional Spaces
In this paper, we study the matrix representations of compressions of Toeplitz operators to the finite dimensional model spaces H2 BH2, where B is a finite Blaschke product. In particular, we
Matrix analysis
TLDR
This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.
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