MATH 304Linear AlgebraLecture 22:Eigenvalues and eigenvectors (continued).Characteristic polynomial.Eigenvalues and eigenvectors of a matrixDefinition. Let A be an nn matrix. A number R is called an eigenvalue of the matrix A ifAv = v for a nonzero column vector v Rn .The vector v is called an eigenvector of Abelonging to (or associated with) the eigenvalue .Remarks. Alternative notation:eigenvalue = characteristic value,eigenvector = characteristic vector. The zero vector is never considered aneigenvector.Diagonal matricesLet A be an nn matrix. Then A is diagonal if andonly if vectors e1 , e2 , . . . , en of the standard basisfor Rn are eigenvectors of A.If this is the case, then the diagonal entries of thematrix A are the corresponding eigenvalues:A=O12O...n Aei = i eiEigenspacesLet A be an nn matrix. Let v be an eigenvectorof A belonging to an eigenvalue .Then Av = v = Av = (I )v = (A I )v = 0.Hence v N(A I ), the nullspace of the matrixA I .Conversely, if x N(A I ) then Ax = x.Thus the eigenvectors of A belonging to theeigenvalue are nonzero vectors from N(A I ).Definition. If N(A I ) 6= {0} then it is calledthe eigenspace of the matrix A corresponding tothe eigenvalue .How to find eigenvalues and eigenvectors?Theorem Given a square matrix A and a scalar ,the following statements ...
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