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NONCOMMUTATIVE ALGEBRAIC EQUATIONS AND NONCOMMUTATIVE EIGENVALUE PROBLEM

Mathematical Physics & Probability

Speaker: Albert Schwarz, Mathematics, UC Davis
Location: 693 Kerr
Start time: Tue, May 9 2000, 4:10PM

Noncommutative algebraic equation has the same form as usual algebraic equation, but the unknown and the coefficients are considered as elements of a noncommutative ring . Several years ago Dmitry Fuchs and I proved an analog of Vieta theorem for equations of this kind. For example, if X_1 and X_2 are two matrices obeying matrix quadratic equation X^2 + AX +B =0 and the difference X_1 - X_2 is an invertible matrix, then trX_1+trX_2=-trA, detX_1 detX_2=detB.I'll start with a proof of noncommutative Vieta theorem based on reduction to noncommutative eigenvalue problem, then I'll show how the same technique can be used to prove a general theorem about the structure of perturbative solutions of noncommutative algebraic equations. This theorem, proved in my recent paper, generalizes a theorem, found by Ashieri, Brace, Morariu and Zumino.( These physicists conjectured their result on the basis of computer calculations and later found a proof of it.)