We define the elementary symmetric polynomials. We show that when the characteristic equation of a square matrix is factored, the symmetric polynomials of the eigenvalues of the matrix are the coefficients of the original characteristic equation. This will give us a formula for the determinant and trace of the matrix in terms of the eigenvalues.
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Elementary Symmetric Polynomials and the Characteristic Polynomial
Теги
mike the mathematicianmike dabkowski mathLinear AlgebraMatrix AlgebraMatrix OperationsLinear Algebra SolutionsMatrix Algebra Solutionsproperties of matricescharacteristic equationcharacteristic equation and symmetric functionssymmetric functions of eigenvaluessymmetric functions of eigenvalues coefficients of characteristic equationcharacteristic equation factorizationelementary symmetric polynomialselementary symmetric polynomials eigenvalues
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