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单词 IfAinMnRAndAIsSupertriangularThenAn0
释义

If AMn(R) and A is supertriangular then An=0


theorem: Let R be commutative ring with identityPlanetmathPlanetmath.If an n-square matrix AMatn(R)is supertriangular then An=0.

proof: Find the characteristic polynomialMathworldPlanetmathPlanetmath of A by computing the determinantMathworldPlanetmath of A-tI. The square matrixMathworldPlanetmath A-tI is a triangular matrixMathworldPlanetmath. The determinant of a triangular matrix is the product of the diagonal element of the matrix. Therefore the characteristic polynomial is p(t)=tn and by the Cayley-Hamilton theoremMathworldPlanetmath the matrix A satisfies the polynomialMathworldPlanetmathPlanetmathPlanetmath. That is An=0.
QED

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更新时间:2025/5/4 14:44:02