A Linear Algebra Method for Solving Systems of Algebraic Equations
Kazuko Kuriyama · 数式処理 · 2000
We consider the exact computation of matrix eigenproblems in residue class rings for solving systems of algebraic equations. We construct multiplication tables using a Grobner basis of a zero-dimensional ideal. Then, we analyze the tables by exactly computing their Frobenius normal forms. The derogatoriness and the diagonalizability are determined by the normal forms, and the problem is divided into four cases: (1) nonderogatory and diagonalizable case, (2) nonderogatory and nondiagonalizable case, (3) derogatory and diagonalizable case, (4) derogatory and nondiagonalizable case. Subsequently, we construct common eigenvectors symbolically, and compute all the exact zeros with their multiplicities. The result of empirical implementation is also shown.