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2 .... pO,jA= [A°,1]j = A0,j for j=1,2,.. )and /1,1/. Assuming that the hypothesis is true for /i- 1,j/ and /i,j - 1/ it will be shown that the hypothesis is valid for /i,j/. 25/ and the definition of pi,j we have Ai,J = A1,OAi-I,J+AO,1Ai,j-1 = At,0 pi-l,JA+ A 0'I pi,j-lA = = pi,~ A . 74/. 77/. 7s/ which is analytic in an open set containing the 2-D eigenvalues /Zli,Z2j / for /0,0/
I067-1073. : Extension of Sylvester's theorem to two-dimensional systems. Bull. Acad. Polon. Sci. Set. Sci. Techn. vol. XXX, no. 53-58. : Minimum energy control. Control and Cybernetics, vol. 121-131. : Controllability of M-dimensional systems. : Function of 2-D matrices. : New results in 2-D systems theory, part I : 2-D polynomial matrices, factorization and coprimeness, part ~ : 2-D state-space models realization and the notions of controllability, observability and minimality. Proc. 6, 1977.
Using the formula nI Ai,J = /i,J/~ /nl,n2/. 5 Find A i'j, /i,j/~/nl,n2/, Step 1. 73/ are ~d Step 2. /2 , 0/, /0, -3/ 12,-31. In this case r/zl,z2/ has the f o ~ r/zl,z2/ = rl0z I + r01z 2 + r00 Substituting the 2-D eigenvalues into the equation ij rloZ 1 + r01z 2 + too = ZlZ 2 30 we obtain 2rio+too--O, -3rol+ro0=O and 2r I0-3rO1 + ro0 = ~i f-D< The solution to the equation is rlO ~d Step 3. 72/ we obtain Ai,J -- 2 i - 1 f _ 3 J j - l ( _ 3 A l , 0 + 2A0,I + 6 I ) -- (-,[; :]+ :,]+ I°]) For finding A i'j the algorithm based on Sylvester*s theorem can also be used [7].