Eigenvectors of certain matrices

Extreme eigenvalues of real symmetric Toeplitz matrices

“…If these vectors are eigenvectors, then their associated eigenvalues are called even and odd, respectively. Drawing on results in [3], it was shown in [6] that, given a real symmetric Toeplitz matrix T of order n, there exists an orthonormal basis for IR n , composed of n − n/2 symmetric and n/2 skew-symmetric eigenvectors of T , where α denotes the integral part of α. In the case of simple eigenvalues, this is easy to see from the fact that, if T u = λu, then T (Ju) = λ(Ju), because JT J = T and J 2 = I.…”

Inverse eigenvalue problems for centrosymmetric matrices under a central principal submatrix constraint

“…Centrosymmetric matrices play an important role in areas such as the numerical solution to certain differential equations [1], various engineering problems [5], and the study of some Markov processes [16].…”

Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices

“…An n × n real matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = P A P (A = −P A P). R n×n r (P) (R n×n a (P)) denotes the subspace reflexive (anti-reflexive) matrices with respect to the n × n generalized reflection matrix P. The reflexive and anti-reflexive matrices have practical applications in many areas such as the numerical solution of certain differential equations [1], pattern recognition [6], Markov processes [42], various physical and engineering problems [7] and so on (e.g. [20,32,43]).…”