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A 1⎦ ... 0 a be a Jordan block. The matrix Jn (a) turns out to be cyclic, because the matrix ⎤ ⎡ λ − a −1 0 ... 0 0 0 ⎢ 0 λ − a −1 . . 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ . 0 ⎥ ⎢ 0 0 λ − a 0 0 ⎥ ⎢ ⎥ ⎢ . . . . .. .. .. ⎥ ⎢ .. ⎥ ⎢ ⎣ 0 0 0 . . λ − a −1 0 ⎦ 0 0 0 . . 0 λ − a −1 is alatent. 93) in the form d(λ) = (λ − λ1 )µ1 · · · (λ − λq )µq , where all numbers λi are different. Consider the matrix J = diag{Jµ1 (λ1 ), . . 97) and its accompanying characteristic matrix λIn − J = diag{λIµ1 − Jµ1 (λ1 ), .

0 0 0 . . λ −1 ⎦ dn dn−1 dn−2 . . 96) with the column b = 0 . . 0 1 , we receive the extended matrix ⎡ ⎤ λ −1 0 . . 0 0 0 ⎢ 0 λ −1 . . 0 0 0⎥ ⎢ ⎥ ⎢ .. .. ⎥ . . . ⎢ . . . ⎥ ⎢ ⎥ ⎣ 0 0 0 . . λ −1 0 ⎦ dn dn−1 dn−2 . . d2 λ + d1 1 This matrix is alatent, because it has a minor of n-th order that is equal to (−1)n−1 . 96) is simple, and therefore, the matrix AF is cyclic. 6. By direct calculation we recognise det(λIn − AF ) = λn + d1 λn−1 + . . + dn = d(λ) . According to the properties of simple matrices, we conclude that the whole of invariant polynomials corresponding to the matrix AF is presented by a1 (λ) = a2 (λ) = .

98) is cyclic. 93), thus J = LAF L−1 , where L in general is a complex non-singular matrix. 15 Simple Realisations and Their Structural Stability 1. The triple of matrices a(λ), b(λ), c(λ) of dimensions p × p, p × m, n × p, according to [69] and others, is called a polynomial matrix description (PMD) τ (λ) = (a(λ), b(λ), c(λ)) . 100) The integers n, p, m are the dimension of the PMD. In dependence on the membership of the entries of the matrices a(λ), b(λ), c(λ) to the sets F[λ], R[λ], C[λ], the sets of all PMDs with dimension n, p, m are denoted by Fnpm [λ], Rnpm [λ], Cnpm [λ], respectively.

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Als Deutschlands Daemme brachen by Helmuth Euler


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