Distribution and estimation for eigenvalues of real quaternion matrices

A brief introduction to quaternion matrices and linear algebra and on bounded groups of quaternion matrices

The division algebra of real quaternions, as the only noncommutative normed division real algebra up to isomorphism of normed algebras, is of great importance. In this note, first we present a brief introduction to quaternion matrices and quaternion linear algebra. This, among other things, will help us present the counterpart of a theorem of Herman Auerbach in the setting of quaternions. More ...

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almos...

A mathematically simple method based on denition for computing eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices

In this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to th...

Real symmetric matrices 1 Eigenvalues and eigenvectors

A matrix D is diagonal if all its off-diagonal entries are zero. If D is diagonal, then its eigenvalues are the diagonal entries, and the characteristic polynomial of D is fD(x) = ∏i=1(x−dii), where dii is the (i, i) diagonal entry of D. A matrix A is diagonalisable if there is an invertible matrix Q such that QAQ−1 is diagonal. Note that A and QAQ−1 always have the same eigenvalues and the sam...

Parallel Computation of Eigenvalues of Real Matrices