### Suzana Miodragović

School of Applied Mathematics and Informatics

Josip Juraj Strossmayer University of Osijek

Josip Juraj Strossmayer University of Osijek

### Research Interests

Numerical Linear Algebra, Perturbation Theory

### Degrees

PhD in theoretical mathematics, Department of Mathematics, University of Zagreb , 2014.BSc in Mathematics and Computer Science, Department of Mathematics, University of Osijek, Croatia, 2007.

### Publications

Journal Publications

- I. Kuzmanović Ivičić, S. Miodragović, Perturbation bounds for stable gyroscopic systems in motion about an unstable equilibrium position, Journal of Computational and Applied Mathematics, (2024), prihvaćen za objavljivanjeRelative perturbation bounds for stable linear gyroscopic systems $\displaystyle M \ddot x(t) + G\dot x(t) + K x(t) = 0$ are considered, where $M, K\in\mathbb{R}^{n\times n}$ are symmetric, $M$ is positive definite, $K$ negative definite and $G \in\mathbb{R}^{n\times n}$ is skew-symmetric such that $|G| > k M - k^{-1} K$ for some $k>0$. This means that all of the eigenvalues of the system are \textcolor{red}{purely} imaginary and semi-simple. For this kind of systems, we present an upper bound for the relative change in eigenvalues as well as the $\sin\Theta$ type bound for the corresponding eigenvectors under the perturbation of the system matrices. The performance of obtained bounds is illustrated in numerical experiments.
- S. Miodragović, N. Truhar, I. Kuzmanović Ivičić, Relative perturbation $\tan \Theta$ theorems for definite matrix pairs, Electronic Transactions on Numerical Analysis
**60**(2024), 364-380 - J. Moro, S. Miodragović, F. de Teran, N. Truhar, Frequency isolation for gyroscopic systems via hyperbolic quadratic eigenvalue problems, Mechanical Systems and Signal Processing
**184**/109688 (2023), 1-19The solutions of a forced gyroscopic system of ODEs may undergo large oscillations whenever some eigenvalues of the corresponding quadratic eigenvalue problem (QEP) $(lambda^2 M + lambda G+K)v=0,quad 0 neq v in mathbb{C}^n,$ are close to the frequency of the external force (both $M,K$ are symmetric, $M$ is positive definite, $K$ is definite and $G$ is skew-symmetric). This is the phenomenon of {colr the} so-called resonance. One way to avoid resonance is to modify some (or all) of the coefficient matrices, $M$, $G$, and $K in mathbb{R}^{ntimes n}$ in such a way that the new system has no eigenvalues close to these frequencies. This is known as the frequency isolation problem. In this paper we present frequency isolation algorithms for tridiagonal systems in which only the gyroscopic term $G$ is modified. To derive these algorithms, the real gyroscopic QEP is first transformed into a complex hyperbolic one, which allows to translate many of the ideas in textcolor{red}{[Mech. Syst. Signal Process., 75:11-26, 2016]} for undamped systems into the full quadratic framework. Some numerical experiments are presented. - I. Kuzmanović Ivičić, S. Miodragović, Perturbation bounds for stable gyroscopic systems, BIT Numerical Mathematics
**63**/1 (2023), 1-16In this paper we consider linear gyroscopic mechanical systems. More precisely, we consider the perturbation theory for stable gyroscopic systems. We present new relative perturbation bounds for the eigenvalues as well as the bounds for the perturbation of the corresponding eigenspaces. Derived bounds are dependent only on system matrices of the original and perturbed systems. The quality of obtained results is illustrated in the numerical experiment. - I. Kuzmanović Ivičić, S. Miodragović, M. Ugrica, The tan Θ theorem for definite matrix pairs, Linear and multilinear algebra
**70**(2022), 1-17In this paper, we consider the perturbation of a Hermitian matrix pair $(H,M)$, where $H$ and $M$ are non-singular and positive definite Hermitian matrices, respectively. A novel upper bound on a tangent of the angles between the eigenspaces of perturbed and unperturbed pairs is derived under a perturbation to the off-diagonal blocks of $H$. The rotation of the eigenspaces under a perturbation is measured in the matrix-dependent scalar product. We show that a $tanTheta$ bound for the standard eigenvalue problem is a special case of our new bound and that the obtained bound can be much sharper than the existing $sinTheta$ bounds. - P. Benner, X. Liang, S. Miodragović, N. Truhar, Relative Perturbation Theory for Quadratic Hermitian Eigenvalue Problem, Linear algebra and its applications
**618**(2021), 97-128In 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. - L. Grubišić, S. Miodragović, N. Truhar, Double angle theorems for definite matrix pairs, Electronic Transactions on Numerical Analysis
**45**(2016), 33-57In this paper we present new double angle theorems for the rotation of the eigenspaces for Hermitian matrix pairs $(H,M)$, where $H$ is a non-singular matrix which can be factorized as $H = G J G^*$, $J = diag(pm 1)$, and $M$ is non-singular. The rotation of the eigenspaces is measured in the matrix dependent scalar product and the bounds belong to the relative perturbation theory. The quality of the new bounds are illustrated in the numerical examples. - D. Jankov Maširević, S. Miodragović, Geometric median in the plane, Elemente der Mathematik
**70**(2015), 21-32 - N. Truhar, S. Miodragović, Relative perturbation theory for denite matrix pairs and hyperbolic eigenvalue problem, Applied Numerical Mathematics
**98**(2015), 106-121In this paper, new relative perturbation bounds for the eigenvalues as well as for the eigensubspaces are developed for definite Hermitian matrix pairs and the quadratic hyperbolic eigenvalue problem. First, we derive relative perturbation bounds for the eigenvalues and the $sin Theta$ type theorems for the eigensubspaces of the definite matrix pairs $(A,B)$, where both $A, B in mathbb{C}^{mtimes m}$ are Hermitian nonsingular matrices with particular emphasis, where $B$ is a diagonal of $pm 1$. Further, we consider the following quadratic hyperbolic eigenvalue problem $(mu^2 M + mu C + K) x =0$, where $M, C, K in mathbb{C}^{ntimes n}$ are given Hermitian matrices. Using proper linearization and new relative perturbation bounds for definite matrix pairs $(A,B)$, we develop corresponding relative perturbation bounds for the eigenvalues and the $sin Theta$ type theorems for the eigensubspaces for the considered quadratic hyperbolic eigenvalue problem. The new bounds are uniform and depend only on matrices $M$, $C$, $K$, perturbations $delta M$, $delta C$ and $delta K$ and standard relative gaps. The quality of new bounds is illustrated through numerical examples. - N. Truhar, L. Grubišić, S. Miodragović, The Rotation of Eigenspaces of Perturbed Matrix Pairs II, Linear and multilinear algebra
**68**/8 (2014), 1010-1031This paper studies the perturbation theory for spectral projections of Hermitian matrix pairs $(H, M)$, where $H$ is a non-singular Hermitian matrix which can be factorized as $H = G J G^*$, $J = diag(pm 1)$, and $M$ is positive definite. The class of allowed perturbations is so restricted that the corresponding perturbed pair $(wtd H, wtd M)= (H+delta H, M+ delta M)$ must have the form $wtd H = wtd G J wtd G^*$, $J = diag(pm 1)$ and $wtd M$ is positive definite. The main contribution of the paper is a $sinTheta$ theorem which generalizes the main result from the first part of the paper to this more general setting. Our estimate, in its most general form, depends on a uniform norm bound on a set of all $J$-unitary matrices which diagonalize $G^*G$. The second main contribution is a new sharp uniform estimate of a norm of all $J$-unitary matrices which diagonalize $G^*G$ such that $H=G^*JG$ is a quasi-definite matrix. The case of a quasi-definite pair is therefore the case where our bounds are most competitive. We present numerical experiments to corroborate the theory.

### Projects

- Accelerated solution of optimal damping problems, — scientific project; supported by the DAAD for period 2020–2021 (investigator ); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany

- Vibration Reduction in Mechanical Systems — scientific project (IP-2019-04-6774, VIMS). This project has been fully supported by Croatian Science Foundation for the period 01.01.2020.–31.12.2023. (investigator)

- Project leader on
*Isolation of the unwanted part of the spectrum in the quadratic eigenvalue problem*. Project*was*funded by J. J. Strossmayer University of Osijek, for period November, 2018.- May, 2020.

- 01.07.2015. – 30.06.2019 investigator on scientific project “
*Optimization of parameter dependent mechanical systems*(IP-2014-09-9540; OptPDMechSys). This project has been fully supported by Croatian Science Foundation

- Mixed Integer Nonlinear Programming (MINLP) for damper optimization — scientific project; supported by the DAAD for period 2015–2016 (investigator); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

- Investigator on Non-linear parameter dependent eigenvalue problem, project leader prof.dr.sc. Ninoslav Truhar. Project
*was*funded by J. J. Strossmayer University of Osijek, for period 10.6.2016.-10.7.2016.

*Optimzacija semi-aktivnog prigušenja kod titrajnih sistema*, project leader: doc. dr. sc. Zoran Tomljanović, Deptartment of Mathematics, University of Osijek. Project*was*funded by J. J. Strossmayer University

### Professional Activities

**Workshop and conference talk
**

- ApplMath2020, 14.-18.9.2020., Brijuni, Croatia

- 28th Biennial Numerical Analysis Conference, Glasgow, 25-28 June 2019

- ICIAM 2019, July, 15-19, Valencia, Spain

**GAMM ANLA Workshop Lund 2018, October, 10-12, 2018, Lund, Sweden**- Eigensubspace perturbation bounds for quadratic eigenvalue problem
**, ApplMath18**, September, 17-20, 2018, Šibenik, Croatia

- International Workshop on Optimal Control of Dynamical Systems and applications, 20-22 June 2018, Osijek, Croatia
- Relative Perturbation Bounds for Hyperbolic Quadratic Eigenvalue Problem, 89th Annual Meeting of GAMM, March 19-23, 2018, Munich, Germany
- Double angle theorems, 6th Croatian MathematicalCongress, June 14-17, 2016, Zagreb, Croatia

- Relative Perturbation Theory for Definite Matrix Pairs, Joint Annual Meeting of GAMM and DMV, March 07-11, Braunschweig, Germany

- Relative Perturbation Theory for Definite Matrix Pairs and Hyperbolic Quadratic Eigenvalue Problem, Workshop on Matrix Equations and Tensor Techniques, September 21-22, Bologna, Italy

- Eigensubspace perturbation theory for definite matrix pairs, Workshop within DAAD project Optimal Damping of Vibrating Systems, Osijek, October 7-10, 2014, Croatia.

- sin 2Θ Theorems for Definite Matrix Pairs,
*10th International Workshop on Accurate Solution of Eigenvalue Problems*, Dubrovnik, June 2 – 5, 2014

- The Rotation of Eigenspace of Perturbed Matrix Pairs, 5th Croation Mathematical Congress, Rijeka, June 18-21, 2012

**Seminars and talks**

- Seminar za optimizaciju i primjene (Odjel za matematiku, Osijek),
*Geometrijski medijan u ravnini*, 2009. - Seminar za Numeričku matematiku i računarstvo (PMF-Matematički odjel, Zagreb),
*Problem**svojstvenih**vrijednosti**hermitskih**matrica**s**“time**reversal**” simetrijom*, 2012. - Topološki seminar, (PMF-Matematički odjel, Zagreb),
*A**-beskona**čno**-prostori**, A**-beskona**čno**-algebre**, A**-beskona**čno**-operadi*, 2012. - Seminar za Numeričku matematiku i računarstvo (PMF-Matematički odjel, Zagreb),
*Rotacija**svojstvenih**potprostora**perturbiranih**matri**čnih**parova*, 2013. - Seminar za optimizaciju i primjene (Odjel za matematiku, Osijek),
*Relativne perturbacijske ocjene ovisne o parametru i hiperbolički svojstveni problem,*, 2014. - Seminar za optimizaciju i primjene (Odjel za matematiku, Osijek),
*Rotacija svojstvenog potprostora perturbiranih definitnih matričnih parova*, 2014. - CSC Seminar (Max Planck Institute, Magdeburg),
*The Rotation of Eigenspace of Perturbed Matrix Pairs*, 2015.

**Participation in summer schools and workshops:**

- The third International School on Model Reduction for Dynamical Control Systems, 5 – 10 October 2015, Dubrovnik, Croatia

- DAAD International School on Linear Optimal Control of Dynamic Systems, 23 – 28 September 2013, Osijek, Croatia

- Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems, October 10-15, 2011, Trogir, Croatia

- DAAD International School “Methods of Numerical Optimization “, May, 2012, Novi Sad/Serbia

- DAAD International School “Hyperbolic Geometry and Arithmetics (Applications of the Selberg Trace Formula)”, October 3-10, 2010, Sarajevo/Bosnia

**Study Visits Abroad**

- 14.02.-28.2.2015., 23.11.-28.11.2015. Max-Planck-Institut für Dynamik komplexer technischer Systeme, Magdeburg
- 2.2.-7.2.2020. University Carlos III, Madrid, Spa

### Teaching

## Teaching