On the distance from a matrix polynomial to matrix polynomials with two prescribed eigenvalues

Document Type : Research Paper


1 Faculty of Science, Yazd University, Yazd, Islamic Republic of Iran.

2 Department of Mathematics, Faculty of Science, Arak University, Arak, Islamic Republic of Iran.

3 Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd, Islamic Republic of Iran.


Consider an n × n matrix polynomial P(λ). A spectral norm distance from P(λ) to the set of n × n matrix polynomials that have a given scalar µ C as a multiple eigenvalue was introduced and obtained by Papathanasiou and Psarrakos. They computed lower and upper bounds for this distance, constructing an associated perturbation of P(λ). In this paper, we extend this result to the case of two given distinct complex numbers µ1 and µ2. First, we compute a lower bound for the spectral norm distance from P(λ) to the set of matrix polynomials that have µ1, µ2 as two eigenvalues. Then we construct an associated perturbation of P(λ) such that the perturbed matrix polynomial has two given scalars µ1 and µ2 in its spectrum. Finally, we derive an upper bound for the distance by the constructed perturbation of P(λ). Numerical examples are provided to illustrate the validity of the method.


[1] J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
[2] J.W. Demmel, On condition numbers and the distance to the nearest ill-posed problem, Numer.Math., 51 (1987),
[3] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.
[4] T. Kaczorek, Polynomial and Rational Matrices: Applications in Dynamical Systems Theory, Springer-Verlag,
London, 2007.
[5] P. Lancaster, Lambda-Matrices and Vibrating Systems, Dover Publications, 2002.
[6] J.M. Gracia, Nearest matrix with two prescribed eigenvalues, Linear Algebra Appl., 401 (2005), 277-294.
[7] R.A. Lippert, Fixing two eigenvalues by a minimal perturbation, Linear Algebra Appl., 406 (2005) 177-200.
[8] A.N. Malyshev, A formula for the 2-norm distance from a matrix to the set of matrices with a multiple eigen-
value, Numer. Math., 83 (1999) 443-454.
[9] A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math., Society, Provi-
dence, RI, Translations of Mathematical Monographs, Vol. 71, 1988.
[10] J. Nocedal, S.J. Wright, Numerical Optimization, second edition, Springer Series in Operation Research and
Financial Engineering, 2006.
[11] N. Papathanasiou, P. Psarrakos, The distance from a matrix polynomial to matrix polynomials with a prescribed
multiple eigenvalue, Linear Algebra Appl., 429 (2008), 1453-1477.
[12] A. Ruhe, Properties of a matrix with a very ill-conditioned eigenproblem, Numer. Math., 15 (1970), 57–60.
[13] J.H. Wilkinson, The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 1965.
[14] J.H. Wilkinson, Note on matrices with a very ill-conditioned eigenproblem, Numer. Math., 19 (1972), 175–178.
[15] J.H. Wilkinson, On neighbouring matrices with quadratic elementary divisors, Numer. Math., 44 (1984), 1–21.
[16] J.H. Wilkinson, Sensitivity of eigenvalues, Util. Math., 25 (1984), 5–76.
[17] J.H. Wilkinson, Sensitivity of eigenvalues II, Util. Math., 30 (1986), 243–286.