Odjel za matematiku

Ninoslav Truhar (Google Scholar Profile) 


Truhar Full Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
phone: +385-31-224-817
fax: +385-31-224-801
email: ntruhar @ mathos.hr
office: 1st  floor


Research Interests

  • Numerical Linear Algebra
  • Systems and Control  Theory
  • Applied Mathematics
Linear Matrix Equations, Linear Vibrating Systems, Damping Optimization, Matrix Perturbation Theory, Perturbation Theory of Invariant Subspaces


  • B. S. in Mathematics and Physics 1987, University of Osijek
  • M. S. In Mathematics 1995, University of Zagreb
  • Ph.D. in Mathematics 2000, University of Zagreb


Study Visits Abroad and Professional Improvement

  • 1997 10-12, visiting researher at The Pennsylvania State University,  State College, PA, USA
  • 1999–2001 post-Ph. D. research at FernUniversitat in Hagen, Germany
  • 2003 guest professor at FernUniversitat in Hagen, Germany (one month)
  • 2004 guest professor at FernUniversitat in Hagen, Germany (one month)
  • 2006 visiting researher at Department of Mathematics, University of Kentucky,
    Lexington, Kentucky, USA
  • 2007 visiting professor at Department of Mathematics at the University of Texas
    at Arlington, Arlington, Texas, USA (one semester)
  • 2013 visiting professor at Department of Mathematics at the University of Texas
    at Arlington, Arlington, Texas, USA (one semester)



Journal Publications

  1. N. Truhar, R. Li, On an Eigenvector-Dependent Nonlinear Eigenvalue Problem from the Perspective of Relative Perturbation Theory, Journal of Computational and Applied Mathematics, 395 (2021)
    We are concerned with the eigenvector-dependent nonlinear eigenvalue problem (NEPv) $H(V)V = V Lambda$, where $H(V) in bbC^{n times n}$ is a Hermitian matrix-valued function of $V in bbC^{n times k}$ with orthonormal columns, i.e., $V^{HH} V = I_k$, $k leq n$ (usually $k ll n$). Sufficient conditions on the solvability and solution uniqueness of NEPv are obtained, based on the well-known results from the relative perturbation theory. These results are complementary to recent ones in [Cai, Zhang, Bai, and Li, {em SIAM J. Matrix Anal. Appl.}, 39:2 (2018), pp.1360--1382], where, among others, one can find conditions for the solvability and solution uniqueness of NEPv, based on the well-known results from the absolute perturbation theory. Although the absolute perturbation theory is more versatile in applications, there are cases where the relative perturbation theory produces better results.
  2. P. Benner, X. Liang, S. Miodragović, N. Truhar, Relative Perturbation Theory for Quadratic Hermitian Eigenvalue Problem, Linear algebra and its applications 618 (2021), 97-128
    In this paper, we derive new relative perturbation bounds for eigenvectors and eigenvalues for regular quadratic eigenvalue problems of the form $(lambda^2 M + lambda C + K)x = 0$, where $M$ and $K$ are nonsingular Hermitian matrices and $C$ is a general Hermitian matrix. These results are based on new relative perturbation bounds for an equivalent regular Hermitian matrix pair $A-lambda B$. The new bounds can be applied to quadratic eigenvalue problems appearing in many relevant applications, such as mechanical models with indefinite damping. The quality of our bounds is demonstrated by several numerical experiments.
  3. N. Truhar, Z. Tomljanović, R. Li, Perturbation Theory for Hermitian Quadratic Eigenvalue Problem -- Damped and Simultaneously Diagonalizable Systems, Applied mathematics and computation 371 (2020)
    The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems $(lambda^2 M + lambda D + K) x=0$. We propose a new concept without linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference $left| | X_2^* M widetilde{;X};_1 |_F^2 - | X_2^* M {;X};_1 |_F^2 right|$, and for $| X_2^* M widetilde X_1 - X_2^* M X_1|_F$, where the columns of $X_1=[x_1, ldots, x_k]$ and $X_2=[x_{;k+1};, ldots, x_n]$ are linearly independent right eigenvectors and $M$ is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for $|sin{;Theta(mathcal{;X};_1, widetilde{;mathcal{;X};};_1)}; |_F$, where $Theta$ is a matrix of canonical angles between the eigensubspaces $mathcal{;X};_1 $ and $widetilde{;mathcal{;X};};_1$, $mathcal{;X};_1 $ is spanned by the set of linearly independent right eigenvectors of SDQEP and $widetilde{;mathcal{;X};};_1$ is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.
  4. N. Truhar, A Note on an Upper and a Lower Bound on Sines between Eigenspaces for regular Hermitian Matrix Pairs, Journal of Computational and Applied Mathematics, 358 (2019), 374-384
    The main results of the paper are un upper and a lower bound for the Frobenius norm of the matrix $sin Theta$, of the sines of the canonical angles between unperturbed and perturbed eigenspaces of a regular generalized Hermitian eigenvalue problem $A x = lambda B x$ where $A$ and $B$ are Hermitian $n times n$ matrices, under a feasible non-Hermitian perturbation. As one application of the obtained bounds we present the corresponding upper and the lower bounds for eigenspaces of a matrix pair $(A,B)$ obtained by a linearization of regular quadratic eigenvalue problem $left( lambda^2 M + lambda D + K right) u = 0 $, where $M$ is positive definite and $D$ and $K$ are semidefinite. We also apply obtained upper and lower bounds to the important problem which considers the influence of adding a damping on mechanical systems. The new results show that for certain additional damping the upper bound can be too pessimistic, but the lower bound can reflect a behaviour of considered eigenspaces properly. The obtained results have been illustrated with several numerical examples.
  5. N. Truhar, Z. Tomljanović, M. Puvača, Approximation of damped quadratic eigenvalue problem by dimension reduction, Applied mathematics and computation 347 (2019), 40-53
    This paper presents an approach to the efficient calculation of all or just one important part of the eigenvalues of the parameter dependent quadratic eigenvalue problem $(lambda^2(mathbf{;v};) M + lambda(mathbf{;v};) D(mathbf{;v};) + K) x(mathbf{;v};) = 0$, where $M, K$ are positive definite Hermitian $ntimes n$ matrices and $D(mathbf{;v};)$ is an $ntimes n$ Hermitian semidefinite matrix which depends on a damping parameter vector $mathbf{;v};= begin{;bmatrix}; v_1 & ldots & v_k end{;bmatrix};in mathbb{;R};_+^k$. With the new approach one can efficiently (and accurately enough) calculate all (or just part of the) eigenvalues even for the case when the parameters $v_i$, which in this paper represent damping viscosities, are of the modest magnitude. Moreover, we derive two types of approximations with corresponding error bounds. The quality of error bounds as well as the performance of the achieved eigenvalue tracking are illustrated in several numerical experiments.




  • Mixed Integer Nonlinear Programming (MINLP) for damper optimization--scientific project; supported by the DAAD for period 2015--2016 (Project director); partner institution: Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg
  • European Model Reduction Network (EU-MORNET). Funded by: COST (European Cooperation in Science and Technology).

         Partner: researchers in model order reduction from 17 countries.

Project run 01/01/2013 - 12/31/2014 founded by DAAD in collaboration between Max Planck Institute for Dynamics Complex Technical Systems Magdeburg, Computational Methods in Systems and Control Theory, Magdeburg, Germany and Department of Mathematics, University of Osijek, Osijek, Croatia   

  • Solution of large-scale Lyapunov Differential Equations,  

    Funded by: FWF Austrian Science Fundation,  FWF project id: P27926
    Researchers: Dr. Hermann Mena (project director, University of Innsbruck, Innsbruck, Austria); Prof. Dr. Alexander Ostermann (University of Innsbruck, Innsbruck, Austria)
    Partners: Universidad Jaime I, Castellon (Spain), University of Tuebingen, (Germany),
    Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg (Germany), Department of Mathematics, University of Osijek (Croatia)

Editorial Service

Member of Editorial Board:

Forthcoming Meetings

  • ????
Committee Memberships
  •  Member of the Scientific Committee of the 6th Croatian Congress of Mathematics (Zagreb, 2016)   





  • SIAM Journal on Matrix Analysis and Applications (SIMAX)
  • SIAM Journal on Scientific Computing (SISC)
  • Linear Algebra and its Applications (LAA)
  • Numerische Mathematik
  • BIT Numerical Mathematics
  • Mathematical and Computer Modelling (MCM)
  • Applied Mathematics and Computation (AMC)
  • International Journal of Computer Mathematics
  • Journal of Applied Mathematics and Computing (JACM)
  • Journal of Sound and Vibration 
  • International Journal of Systems Science
  • International Journal of Computer Mathematics
  • Numerical Algorithms
  • Central European Journal of Mathematics
  • Bulletin of the Iranian Mathematical Society 
  • Glasnik matematički
  • Mathematical Communications


  • AMS Mathematical Review   (since 2006)
  • Zentralblatt MATH


Service Activities


  • Chairman of Osijek Mathematical Society, 2003--2013 
  • Chairman of the Mathematical Colloquium, 2005-2017


Rezultati pismenog ispita Razlikovna godina Građevinski i arhitektnonski fakultet;

rok 11. 02. 2022:

0149228437 (DŠ) ocjena 2 (53 boda)
0149226175 (MV) ocjena 2 (40 bodova)




Konzultacije (Office Hours): Utorak (Tue)  11:30am-12:15pm, Srijeda (Wed) 11:30am-12:15pm. Konzultacije su moguće i po dogovoru.


Dodiplomska nastava:


 Diplomska nastava:


Teme za diplomske radove: popis tema







  • Birthdate: May 4, 1963
  • Birthplace: Osijek, Croatia
  • Citizenship: Croatian
  • Family: Married



I am a fan and supporter of basketball club KK Vrijednosnice Osijek