16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-21To understand the notion of eigenvector better, observe that the equationAξ = rξis equivalent to either of the equationsor(rI − A)ξ = 0(A − rI)ξ = 0where I is the n × n identity matrix.If either of these equations had a non-zero vector ξ as a solution, then itwould follow thatdet(rI − A) = 0 (10)The expression det(rI − A) is actually a polynomial of degree n of theformz(r) = r n + a n−1 r n−1 + . . . + a 0and the above equation can be written as z(r) = 0.The polynomial z(r) is called the characteristic polynomial of the matrixA, and its roots are the eigenvalues of A.The existence of these roots is provided by theTheorem (Fundamental Theorem of Algebra) Letp(r) = a n r n + a n−1 r n−1 + . . . + a 0be a polynomial of degree n (i.e. a n ≠ 0) with complex coefficientsa 0 , a 1 , . . . , a n .Then, there is a complex number α such that z(α) = 0.Remark. The Euclidean algorithm for positive integers states that giventwo positive integers p > q there are integers k > 0 and 0 ≤ s < q such thatp = kq + sThere is an analogous result for polynomials. Let us denote by deg(z(r))the degree of the polynomial z(r) (with real of complex coefficients).