By Marvin L. Minsky
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33 PROOF: A ' z (°) = 0 implies w(°)*6 ' = [0, implies that _(o)'~. , 0] = 0. A'z(i) = B'z (i-~} ~ ( J - ' " ~' - ~(J)" B' = [~J-~)" A + ~i-1)" C ' Q C + z~~ - ' " s * c - z~j-~)" E, - ~ J - " E " + z~J)'A* + z(J)'B', ~ - ~ ) ' B + z~i-')'c's + ~J-')'RI = o by using successive equations for j , j - 1. )'o + ~ ) ' s ' c , - z ? ) ' E ' , 4~-1)'B + . )'RI = o. 33 P~EMARK. a) It follows that to every Kronecker block to an eigenpair (o~,fi) ~ (fl, ~), c~ ~ 0 there exists a Kroneeker block of the same size to (3, ~ ) b) If a A ' - fiB' is a real pencil, then pairing is for blocks to (c~, fl) ¢ (fl, ~ ) , where c~ ¢ ft.
2). 3) aZ: - f l . )c [ -E" 0 S*C 0 n [i - fl BR-1B" i] -A* + C*SR-~B • -B" respectively. 6) . - f BR-'s'c -BR-1B* (A--BR-IS*C) ] * ' respectively, are regular. These axe the forms in which the pencils usually appear in the literature, since in many applications R is positive definite. g. Emami-Naeini [E 5] for a list of references. 6) are very well analyzed concerning their eigenstructure since Z - fl M is a Hamiltonian pencil and c~ £' - / 7 M ' is a symplectic pencil. See for example Laub [L 11], [L 12], Laub/Meyer , Lee [L 17], Paige/Van Loan [P 2], Pappas et al [P 5], Singer/Hammarling [S 10] or in a more general setting Gohberg et al [C 14], Lancaster/Rodman [L 4], Mehrmann [M 8].
Then there exists a unique Q E / 4 S 2 . e. R = 1 R~l ' positive diagonM elements. g. 16.
Computation, Finite And Infinite Machines by Marvin L. Minsky