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Determinants of the Block Arrowhead Matrices

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Determinants of the Block Arrowhead Matrices Determinants of the block arrowhead matrices Edyta HETMANIOK, Micha lRO´ ZA˙ NSKI,´ Damian SLOTA, Marcin SZWEDA, Tomasz TRAWINSKI´ and Roman WITULA Abstract. The paper is devoted to the considerations on determinants of the block arrowhead matrices. At first we discuss, in the wide technical aspect, the motivation of undertaking these investigations. Next we present the main theorem concerning the formulas describing the determinants of the block arrowhead matrices. We discuss also the application of these formulas by analyzing many specific examples. At the end of the paper we make an attempt to test the obtained formulas for the determinants of the block arrowhead matrices, but in case of replacing the standard inverses of the matrices by the Drazin inverses. Keywords: determinant, block matrix, arrowhead matrix, Drazin inverse. 2010 Mathematics Subject Classification: 15A15, 15A23, 15A09. 1. Technical introduction – motivation for discussion Observing the mathematical models, formulated for describing various dynami- cal systems used in the present engineering, one can notice that in many of them the block arrowhead matrices occur. One can find such matrices in the models of telecommunication systems [10], in robotics [18], in electrotechnics [17] and in auto- matic control [21]. In robotics, while modelling the dynamics of the kinematic chains of robots, or in electrotechnical problems, while modelling the electromechanical con- verters, there is often a need to formulate some equations in the coordinate systems different the ones in which the original model was formulated. Especially in the theory E. Hetmaniok, M. R´o˙za´nski, D. S lota, M. Szweda, R. Witu la Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: {edyta.hetmaniok, michal.rozanski, damian.slota, marcin.szweda, roman.witula}@polsl.pl T. Trawi´nski Department of Mechatronics, Silesian University of Technology, Akademicka 10A, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witu la, B. Bajorska-Harapi´nska, E. Hetmaniok, D. S lota, T. Trawi´nski (eds.), Selected Problems on Experimental Mathematics. Wydawnictwo Politechniki Sl¸askiej,´ Gliwice 2017, pp. 73–88. 74 E. Hetmaniok, M. R´o˙za´nski, D. S lota, M. Szweda, T. Trawi´nski and R. Witu la of electromechanical converters one applies very widely the possibility of transform- ing the mathematical models of the electromechanical converters form the natural coordinate systems into the biaxial coordinate systems. The main goal of these trans- formations is to eliminate the time dependence occurring in the coefficients of mutual inductances, to increase the number of zero elements occurring in the matrices of the appropriate mathematical model and, in consequence, to simplify this model and to shorten the time needed for its solution. The forms of matrices transforming the variables, occurring in the electromechanical converters, from the natural coordinate systems into the biaxial coordinate systems can be deduced on the way of physical reasoning. These matrices can be also found by using the methods of determining the eigenvalues. Formulated matrices are applied, among others in the Park, Clark and Stanley transformations [12, 9]. Elements of the transformation matrices contain the eigenvalues of matrices of the electromechanical converter inductance coefficients. These matrices can have, for example, the following forms [7]: cos ϑ 1 cos ϑ 2 cos ϑ 3 2 s s s [Ks]= sin ϑs1 sin ϑs2 sin ϑs3 , (1) 3 1 1 1 r √2 √2 √2 1 cos αr cos2αr ... cos(Qr − 1)αr 0 − sin αr − sin 2αr ... − sin(Qr − 1)αr 2 1 cos2αr cos4αr ... cos2(Qr − 1)αr [Kr]= 0 − sin 2αr − sin 4αr ... − sin 2(Qr − 1)αr , (2) Qr r . . 1 1 1 1 1 √2 √2 √2 √2 √2 where t ϑs1 = ϑs10 + Ωx dt, (3) Z0 t t 2π ϑ 2 = ϑ 20 + Ω dt = ϑ 10 + + Ω dt, (4) s s x s 3 x Z0 Z0 t t 4π ϑ 3 = ϑ 30 + Ω dt = ϑ 10 + + Ω dt, (5) s s x s 3 x Z0 Z0 whereas ϑs10, ϑs20, ϑs30 denote the initial angles between the axes of respective phases of the stator and axis X of the biaxial system XY for moment of time t = 0; p means the number of pole pairs, αr denotes the rotor bar pitch and Ωx describes the angular velocity of rotation of the biaxial system XY around the stator. Matrix of the mutual and self inductance coefficients written in the natural coor- dinate system, that is in the phase system, possesses the block structure – it is the full matrix of the form Mss Msr M = T . (6) M Mrr sr Particular forms of matrices, of this type can be found, for example, in [7]. Determinants of the block arrowhead matrices 75 Method of transforming the matrix (6) of inductance coefficients into the new coordinate system XY by using the transformation matrices (1) and (2) is as follows XY T Mss = [Ks][Mss][Ks] , (7) XY T Mrr = [Kr][Mrr][Kr] , (8) XY T Msr = [Ks][Msr][Kr] . (9) By merging the obtained results we get the block arrowhead matrix [8], also in the form presented in paper [17]. Matrices (1) and (2) are often used for constructing the systems for regulating the electromechanical converters, such as the various versions of the vector control methods for the squirrel cage induction motors [13, 19]. Thus, there is a need for de- veloping the effective methods of calculating the determinants of the block arrowhead matrices. Only to this issue the second part of this paper (see [17]) will be devoted. 2. The block arrowhead matrices We use the following notation in this paper: ¼ denotes the zero matrix of the respective size (of dimensions m × n), ½ denotes the identity matrix of the respective order (of order n). We present now the main result of this paper concerning the form of determinants of the block arrowhead matrices. In Section 3 we examine few examples illustrating the application of the obtained relations. Theorem 2.1. Aa a Ba b 1. Let us consider the block matrix M2 = × × . The following implications Cb a Db b hold: × × 1 (a) If det D 6=0, then det M2 = det D det(A − BD− C). 1 (b) If det A 6=0, then det M2 = det A det(D − CA− B). In Remark 2.2, after the proof of the above theorem, we discuss the omitted situation when det A = det D =0 and a 6= b. Aa a Ba b Ea c × × × 2. Let us consider the block matrix M3 = Cb a Db b ¼b c . The following impli- × × × Fc a ¼c b Gc c × × × cations hold 1 (a) If det A 6=0 and det(D − CA− B) 6=0, then 1 1 det M3 = det A det(D − CA− B) det(G − F A− E − 1 1 1 1 − F A− B(D − CA− B)− CA− E). (10) 76 E. Hetmaniok, M. R´o˙za´nski, D. S lota, M. Szweda, T. Trawi´nski and R. Witu la 1 We note that if also det D 6=0, then for the inverse of D −CA− B the following Banachiewicz formula (see [1]), also called the Sherman-Morrison-Woodbury (SMW) formula (for some more historical information see [22, subchapter 0.8]), could be applied (see also S. Jose, K. C. Sivakumar, Moore-Penrose inverse of perturbated operators on Hilbert Spaces, 119-131, in [2]): 1 1 1 1 1 1 1 (D − CA− B)− = D− + D− C(A − BD− C)− BD− . 1 The matrix A − BD− C is of order a × a and the above formula is useful in situations when a is much smaller than b and in all other situations when certain 1 structural properties of D are much more simple than of D − CA− B. (b) If det D 6=0 and det G 6=0, then 1 1 det M3 = det D det G det(A − BD− C − EG− F ). (11) B E (c) If either rank + rank ≤ b + c − 1 or rank C D + rank F G ≤ D G b + c − 1, then det M3 =0. 1 (d) If det G 6=0, det(A − EG− F ) 6=0 and D = ¼, then 1 1 1 det M3 = det G det(A − EG− F ) det(−C(A − EG− F )− B). (12) In the sequel, if blocks B and C of M3 are the square matrices (that is a = b) and D = ¼, then a det M3 = (−1) det G det B det C. (13) At last, if A − EG 1F B A − EG 1F rank − = rank − + rank B C ¼ C 1 = rank C + rank A − EG− F B <a + b, then det M3 =0. 3. The determinant of the following arrowhead matrix (A11)b1 b1 (A12)b1 b2 (A13)b1 b3 ... (A1n)b1 bn × × × × ¼ (A21)b2 b1 (A22)b2 b2 ¼b2 b3 ... b2 bn × × × × ¼ (A31)b3 b1 ¼b3 b2 (A33)b3 b3 ... b3 bn Mn = × × × × (14) . . ¼ (An1)bn b1 ¼bn b2 bn b3 ... (Ann)bn bn × × × × is equal to (a) n n 1 det Mn = det A11 − A1iAii− Ai1 det Ajj , (15) i=2 ! j=2 X Y whenever det Aii 6=0 for i =2, 3,...,n, Determinants of the block arrowhead matrices 77 (b) n bi det Mn = (−1) det Ai1 det A1i det Ajj , (16) j=2 jY=i 6 whenever there exists i ∈{2, 3,...,n} such that Aii = ¼ and bi = b1. 4. The determinant of arrowhead matrix (14) is equal to n i 1 1 − 1 − det Mn = det A11 det Aii − Ai1 A11 − A1j Ajj− Aj1 A1i , (17) i=2 j=2 Y X i 1 − 1 whenever det Ajj 6=0 for each j =1, 2,...,n−1 and det A11− A1j Ajj− Aj1 6=0 j=2 for each i =3, 4,...,n − 1.
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