892 research outputs found
Procrustes problem for the inverse eigenvalue problem of normal (skew) -Hamiltonian matrices and normal -symplectic matrices
A square complex matrix is called (skew) -Hamiltonian if is
(skew) hermitian where is a real normal matrix such that , where
is the identity matrix. In this paper, we solve the Procrustes problem to
find normal (skew) -Hamiltonian solutions for the inverse eigenvalue
problem. In addition, a similar problem is investigated for normal
-symplectic matrices.Comment: 25 page
On the probability that all eigenvalues of Gaussian, Wishart, and double Wishart random matrices lie within an interval
We derive the probability that all eigenvalues of a random matrix lie
within an arbitrary interval ,
, when is a real or complex finite dimensional Wishart,
double Wishart, or Gaussian symmetric/Hermitian matrix. We give efficient
recursive formulas allowing the exact evaluation of for Wishart
matrices, even with large number of variates and degrees of freedom. We also
prove that the probability that all eigenvalues are within the limiting
spectral support (given by the Mar{\v{c}}enko-Pastur or the semicircle laws)
tends for large dimensions to the universal values and for
the real and complex cases, respectively. Applications include improved bounds
for the probability that a Gaussian measurement matrix has a given restricted
isometry constant in compressed sensing.Comment: IEEE Transactions on Information Theory, 201
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