By Deane Yang
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Extra resources for Involutive Hyperbolic Differential Systems
21) holds. Now consider sf2 7^ 0, s^ = s. The first remark is that any symbol with these reduced Cartan characters is involutive. Moreover, such a symbol has a normal form which is given by the following classical theorem in linear algebra. A pencil of matrices is a two-dimensional subspace in the vector space of all matrices of a given dimension. 22) PROPOSlTlON(Kronecker pencil lemma). 23) LCl 0 ... 0 0 0 L^£o 2 ... 0 0 at + brj = 0 0 Uk V 0 0 0 0 A( + Br} J where Lt is an e-by-e + 1 matrix of the following form / Z r) 0 .
If a linear partial differential operator is symmetrizable hyperbolic, then its symbol at each point in the domain is determined hyperbolic. The converse is not quite true because the symmetrizing function r must also be a smooth function of the domain over which the differential operator acts. On the other hand, the differential operator is strictly hyperbolic if and only if at each point in the domain, the characteristic variety of the symbol of the operator is strictly hyperbolic. (2) Strict hyperbolicity implies determined hyperbolicity.
22) PROPOSlTlON(Kronecker pencil lemma). 23) LCl 0 ... 0 0 0 L^£o 2 ... 0 0 at + brj = 0 0 Uk V 0 0 0 0 A( + Br} J where Lt is an e-by-e + 1 matrix of the following form / Z r) 0 . . £ and A£ + Brj is a pencil of square matrices. A proof of the proposition may be found in . Let 6 be the size of A£ + Brj. Then ci, — , e*, 6 are invariants of eg -(- ^7 ^ d therefore of a. They satisfy: 26 Deane Yang k k 1=1 Therefore, k = s'2. A modern restatement of the Kronecker pencil lemma is known as the Grothendieck's theorem on holomorphic vector bundles over the projective line.
Involutive Hyperbolic Differential Systems by Deane Yang