On Approximation Problems With Zero-Trace Matrices

@article{Zietak1996OnAP,
  title={On Approximation Problems With Zero-Trace Matrices},
  author={Krystyna Zietak},
  journal={Linear Algebra and its Applications},
  year={1996},
  volume={247},
  pages={169-183}
}
  • K. Zietak
  • Published 1 November 1996
  • Mathematics
  • Linear Algebra and its Applications
Abstract We consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms. We deal with special cases of approximation of a matrix by zero-trace matrices. Moreover, some characterizations of zero-trace matrices are given by means of matrix approximation problems. 
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References

SHOWING 1-10 OF 20 REFERENCES
Properties of linear approximations of matrices in the spectral norm
Abstract The properties of linear approximations of a matrix are presented with respect to the spectrum norm. We describe the connection between all spectral approximations of a given matrix, and weExpand
On zero-trace matrices
Let A be an n × n complex matrix, and let 1 ⩽ p < ∞. It is shown that tr A = 0 if and only if ‖I + zA‖p ⩾ nsol1p for all zϵC. Some related characterizations are also given.
Characterizations of the trace
Abstract The paper contains a number of equivalent conditions which characterize the trace among the linear functionals on the matrix algebra. Some of these results are extended to more generalExpand
Subdifferentials, faces, and dual matrices
Abstract A characterization of the dual matrices for the unitarily invariant norms is given. Moreover, the connection between the dual matrices, the subdifferentials of matrix norms, and the faces ofExpand
On the characterization of the extremal points of the unit sphere of matrices
Abstract We give a characterization of the extremal points of the unit sphere of matrices for the unitarily invariant norms. We also investigate the properties of the dual matrices. Moreover, we showExpand
Characterization of the subdifferential of some matrix norms
Abstract A characterization is given of the subdifferential of matrix norms from two classes, orthogonally invariant norms and operator (or subordinate) norms. Specific results are derived for someExpand
Topics in Matrix Analysis
1. The field of values 2. Stable matrices and inertia 3. Singular value inequalities 4. Matrix equations and Kronecker products 5. Hadamard products 6. Matrices and functions.
A Continuous Selection of the Metric Projection in Matrix Spaces
We denote by C m×n the vector space of complex m×n matrices over C, m, n∈N, with elements A, B,.... For A ∈ C m×l and B ∈ C l×n, l, m, n∈N, AB denotes the matrix product of A and B in C m×n, A* theExpand
Matrix Perturbation Theory
$$X = {X_0} + \varepsilon {X_1} + {\varepsilon ^2}{X_2} + \cdots $$ (1.1.1) which acts in the n-dimensional (complex) vector space R.
Exposed faces and duality for symmetric and unitarily invariant norms
Abstract Let ψ be a unitarily invariant norm on the space of (real or complex) n×m matrices, and g the associated symmetric gauge function: thus ψ(A)g(s(A)), where s(A) is the decreasing sequence ofExpand
...
1
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