Diagonal canonical form of interval matrices and applications on dynamical systems

Finding the simplest form of a set of quantities is an important aspect of any branch of Mathematics. Of course, the simplest form or the canonical form as we often call it in mathematics, must possess all the important characteristics of the set of quantities. A real square matrix satisfying certain conditions can be brought to diagonal form which is its simplest form such that the diagonal form retains the eigenvalues, determinants, trace, rank, nullity,.. of the original matrix. Many computations with matrices become easier if one can diagonalize the matrices. In this article, we suggest an approach for diagonalizing interval matrices employing a novel methodology called the pairing technique, which will make it simpler and more effective to classify and investigate interval matrices. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. We also discuss two real world applications on planar systems and linear discrete dynamical systems.


Introduction
In mathematics, the simplest form or the canonical form of a set of quantities must possess all the important characteristics of the set of quantities. For example, a second degree equation in x and y under certain conditions represents an ellipse but the canonical equation of an ellipsis is + = 1 for a suitable choice of x and y -axes. Similarly, a real square matrix that meets specific requirements can be reduced to diagonal form, which is the simplest form possible while still preserving the original matrix eigenvalues, determinants, trace, rank, nullity, etc.
In operator theory, the study of partial differential equations is easy if the operator is diagonal with respect to the basis.
An uncoupled system of differential equations represents the matrix which is diagonal and hence the general solution of the system is easy to find.
We observe that in matrix theory over , diagonalization of an (n × n) matrix ( ) Î  A M n is possible if and only if the matrix 'A' has n-real eigenvalues and n-linearly independent eigenvectors in  n or the minimal polynomial m(t) of A is square free or  n has a basis in which each vector is an eigenvector of A or there exist an invertible matrix P such that P −1 AP is diagonal.
Diagonalization of matrices is widely applied in computing the general n-th power and inverse of matrices, to solve linear systems of differential equations and systems of difference equations etc. It is applied to determine the steady states of linear and non-linear systems. Also it is used to determine the definiteness of a quadratic form by examining the eigenvalues. Many more similar applications of diagonalization are possible.
In real life situations, interval matrices occur in a natural way when one or more entries are imprecise and are expressed as closed and bounded intervals in . n n is the set of all interval matrices and E is the field isomorphic to  obtained through an equivalence relation on . Hence E n is a vector space over E and M n (E) the set of all (n × n) matrices with entries in E is a vector space over E of dimension n 2 . An interval matrix in  is an element of an equivalence class in M n (E) and the results obtained for M n (E) are passed on to interval matrices by an appropriate pairing technique [1]. All these results are proved with consummate ease for matrices in M n (E) in view of the isomorphism of E and .
The process of diagonalizing an interval matrix, if it is amenable to diagonalization can be defined and obtained depending on an ordered basis or independent of the basis for the vector space M n (E) over E. The study of diagonal form of interval matrices will help us to classify and study interval matrices more effectively and more easily.
Hansen and Smith [2,3] initiated the importance of interval arithmetic in matrix computations. Motivated by this, many authors such as Alefeld and Herzberger [4], Luc Jaulin et al [5], Neumaier [6], Assem Deif [7], Ganesan and Veeramani [8,9], Nicolas Goze [10], Moore et al [11], Sengupta and Pal [12] etc have studied interval matrices. Several authors such as Pastravanu et al [13,14], Berman et al [15], Kaszkurewicz et al [16] etc studied the diagonal stability of interval matrices and their applications. A new class of arithmetic operations on interval numbers was introduced by Ganesan and Veeramani [8]. Ganesan [9] described several significant interval matrix features. The singularity property was used by Abhirup Sit [17] to study the eigenvalues of interval matrices. Assem Deif [7] investigated interval eigenvalue problems. This result was obtained by using the interval extension of the single step eigen perturbation method. The interval matrix's deviation amplitude is regarded as a perturbation around the nominal value of the interval matrix. Hema Surya et al [1] defined a field and a vector space over a field including equivalence classes to explain interval linear algebra in a sound algebraic environment. Pastravanu et al [13,14] discussed the diagonal stability of interval matrices and their applications in linear dynamical systems. Berman et al [15] have also discussed the matrix diagonal stability and its implications.Using Montroll's traditional method, Nathali et al [18] examined data on population for Peru from 1961 to 2013. Three types of growth rates are taken into account to simulate population dynamics involving real numbers, triangle type-1 and interval type-2 fuzzy parameters.
According to Malthus law of population theory [19], rising population rates lead to greater supply of labor which necessarily drives down wages. Malthus simply believed that poverty is a natural byproduct of the growth of populations. Barros et al [20] analyzed the behavior of models which describe the population dynamics under uncertain situations. Also this model has demographic and environmental ambiguity. Khan et al [21], Mazenc and Bernard [22] and Sergiyenko et al [23] developed the design and applications of interval observers for uncertain dynamical systems . Also their approach is substantially different from traditional methods for establishing control laws for discrete and continuous time perturbed processes or performing robust stability analysis. Carlos et al [18] rediscovered the Power Iteration Method from the Projected Gradient Method and also establishing Convergence of the discrete dynamical system solution. Ufuktepe et al [19,20] discussed the discrete dynamical systems by using Mathematica software and stability of the one dimensional system, the Cobweb diagram for one dimensional system, the time series diagram, the phase plane diagrams for two-dimensional systems. Tang et al [24] proposed a two step interval estimation method for discrete time linear systems by integrating robust observer design and reachability analysis. Efimov et al [21,22] studied the interval state observer design for time-varying discrete-time systems and they offered three solutions: For a generic timevarying system, a system with positive state, and a particular type of periodic systems and also they reviewed the important tools and techniques for design of interval observers for continuous-time, discrete-time and timedelayed systems. Feeney [25] explained the population dynamics based on birth intervals and parity progression. Khan et al [26] examined the effects of socioeconomic, demographic and proximate determinants on the length of birth intervals of women of Bangladesh.
Naresh et al [23] studied Elzaki transform method for solving the two tank mixing problems. Taoyan et al [24] discussed the simulation results on the water tank level control system. They proposed method which is better static and dynamic control with stronger robust performance than the traditional fuzzy control method. Ukwu et al [25] explained a class of mixing problems as they pertain to scalar delay differential equations. Yongjian et al [26] proposed a time-dependent concentration interval analysis (CIA) method to solve the problems associated with the synthesis of discontinuous or batch water-using systems involving both non-masstransfer-based and mass-transfer-based operation. They showed the possibility of water reuse and the amount of water reused under time constraints for minimizing the consumption of freshwater in single or repeated batch/ discontinuous water-using systems. Julian et al [27] proposed method for determining the minimum water tank volume taking into account all input variables, which might be applied in a simple and useful manner in everyday water supply management. Vrachimis et al [28] discussed a methodology for generating hydraulic state bounding estimates based on interval bounds on the parametric and measurement uncertainties.Nirmaladevi et al [29] discussed population growth, calculating the rate of growth and decay and rate of mixture using differential equations.
In this paper, we propose a method for diagonalizing interval matrices using pairing techniques [1]. We establish some important theorems on diagonalization of interval matrices. We discuss two real life applications on planar systems and linear discrete dynamical systems.

Preliminaries
We recall some basic notions and notation on closed and bounded intervals in  [1]. Let a a a a , : be the set of all closed and bounded intervals. If a L = a U , thenã is a degenerate interval.
The intervals are identified with an ordered pair 〈m, w〉 defined as follows: and henceã can be uniquely expressed as (˜) (˜) á ñ m a w a , . Conversly, 0, 0 , then˜» a 0, but the converse need not be true. If˜» a 0, thenã is said to be a non-zero interval. If (˜) > m a 0 thenã is said to be a positive interval and is denoted by˜ a 0.

Interval matrices
We recall some basic notions and notation on interval matrices [1]. A classical matrix is a rectangular array of elements of a field . When matrix entries are inaccurate or unclear, we use interval matrices, where each entry is We use E m× n to denote the set of all equivalence classes of (m × n) interval matrices. By (˜) m A we denote a matrix whose entries are the corresponding midpoints of the entries ofÃ. i.e) . The width of an interval matrixÃ is the matrix of widths of its interval elements is a interval column vector whose components are closed and bounded intervals. We use  n to denote the set of all n-dimensional interval vectors. By (˜) m x we denote a column vector whose entries are the corresponding midpoints of the entries ofx. i.e) (˜) and the width of interval vector is defined by . An interval vector is said to be non-negative if all of its components are non-neagative.

Definition 1. Determinant of an interval matrixÃ is an interval in  and is defined by
Definition 3. If |˜|» A 0, then the inverse ofÃ exist and is defined bỹ be the interval eigenvalues ofÃ. The spectral radius ofÃ is defined by is an eigen value of (˜)} w A .
Definition 6. Letx be a non-negative interval vector. If sum of all entries of (˜) m x is 1, thenx is called a probability interval vector.
Definition 7. A transition interval matrix or stochastic interval matrix is a square interval matrix whose columns are probability interval vectors.

Diagonalization of interval matrices
Definition 8. An interval matrixÃ is said to be diagonalizable over  ifÃ has n interval eigenvalues and n linearly independent normalized interval eigenvectors over . is linearly independent in E n . Thus (˜) m A has n eigenvalues and n linearly independent normalized eigenvectors over .
Conversely, suppose (˜) m A has n eigenvalues and n linearly independent normalized eigenvectors over . Hence (˜) m A is diagonalizable over . Using pairing techniques [1], the interval matrixÃ has n interval eigenvalues and n linearly independent normalized interval eigenvectors over . Hence,Ã is diagonalizable .
Mimicking the proof of theorem (1), we can prove the following.
Theorem 2. An interval matrixÃ is diagonalizable over  iff the minimal polynomial of (˜) m A is square free.

3.
(˜) 1 and there is an interval eigenvaluel ofÃ with | (˜)| l = m 1 such that the algebraic multiplicity ofl is strictly greater than its geometric multiplicity.

An application to planar system
A typical mixing problem deals with the amount of salt in a mixing tank. Salt and water enter to the tank at a certain rate, they are mixed with what is already in the tank, and the mixture leaves at a certain rate. The multicompartment model is more challenging and requires the use of techniques of linear algebra. Systems of differential equations can be used to model how substances flow back and forth between two or more compartments. In particular, the systems of ordinary differential equations -associated matrix deserve to be studied since it determines the qualitative behavior of the solutions. As shown in figure 1, there are two tanks, A and B, between which a mixture of brine flows. Tank A has 300 litres of water with 100 kg of salt dissolved in it, and Tank B has 300 litres of pure water. At a rate of 500 litres per hour, fresh water is pumped into Tank A, and at a rate of 900 litres per hour, brine is pumped from Tank A into Tank B. Tank B's brine is also pumped back into Tank A at the rate of 400 litres per hour. Tank B's brine is also drained at the rate of 500 litres per hour. Every brine mixture is stirred well. Find the amount of salt in the tanks at any given time t. What happens over a long run?
Solution: Let x = x(t) be the amount of salt in Tank A at time t and y = y(t) be the amount of salt in Tank B at time t.
We know that the salt concentrations in the two tanks are x/300 kg per liter and y/300 kg per liter. Thus, the rate of change of salt concentrations in each tank can be described with a system of differential equations, The matrix form of the system (2) is given by 3 .
. The eigenvalues of A are −1, −5 and the corresponding eigenvectors are ( ) . Since the eigenvalues are real and distinct, the general solution of the given system of interval linear differential equations (2) is x t y t c e c e 2 3 Any sort of water, including pure, fresh, and brine, has some salt in it. It implies that very little salt might be present in pure water also. Therefore, let's assume that initially, the salt concentrations in tanks A and B arẽ 0, 1 respectively. Applying these initial conditions in equation (3), we get  The amount of salt in Tank B at any time t is given bỹ After, a long period of time (i.e. t → ∞ ) the salt content in both tanks A and B will stabilize. Figure 2 depicts the salt content in Tank A and its exponential decrease as a result of water forth to Tank B. Also it shows that the salt content will stabilize after some time. Figure 3 depicts the salt content in Tank B and its exponential rise as a result of water flow from Tank A. Also it shows that the salt content will stabilize after some time.
The combined level of salt content in both tanks (A and B) is shown in figure 4. It demonstrates that the salt levels in both tanks will stabilize over time.
6. An application to linear discrete dynamical interval system Many branches of applied science make use of linear discrete dynamical interval systems, including meteorology, psychology, studies of economics, etc. The relevance of such long-term behaviour to the real world is significant. The (k + 1)-th state of a discrete evolution system can be written as x k+1 = Ax k and its solution is needed in many areas of applied mathematics. The k-th state of a system whose initial state is x 0 is provided by the equation x k = A k x 0 .
Let C and S be a city and its suburb respectively. Assume that each year around 10% of the population of S migrates to C, and around 5% of the population of C moves to S. What is the long term effect of this on the populations of C and S ? Are they likely to stabilize? Solution: The birth and death rates will change every minute. For this situation, the classical method cannot be used successfully. So we model as a discrete evolution interval system due to the imprecision. It is more appropriate for getting outcomes very ease and reliable. Hence the current population of the city C and its suburb S bex 0 andỹ 0 respectively. Then the population interval vector for the current year is given by  and S at the end of first year after the migration will be re-distributed as˜[ Similarly, the population of C and S at the end of the second year will bẽ˜˜˜= » A A p p p .    The population of the city C and its suburb S at the end of the k-th year be˜˜» A p p   respectively. The following figure 5 shows that in the long run, the population of the city C stabilies approximately at double that of its suburb S.

Conclusion
In this paper, we have analyzed theorems for diagonalization of interval matrices corresponding to matrices over . We have extended definitions for real matrices to corresponding definitions for interval matrices. Four important necessary and sufficient conditions for diagonalization of real interval matrices have been established. We have shown that how interval matrix diagonalization can be used to efficiently compute the powers of interval matrices which in turn help us to deal with real-life problems effectively. Also we discussed a mixing problem with two water tanks and population migration problem under interval uncertainty.