(5.1.2)--5.1.2Findtheeigenvaluesofinverse.ppt

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1、Linear AlgebraFind the Eigenvalues of Invertible Matrix and Polynomial MatrixProperty 1.Let the square matrix A be invertible,and is the eigenvalue of A,then 1/is the eigenvalue of A-1.Proof.First prove that 0.Assume,by contradiction,that=0 is the eigenvalue of matrix A,then we obtain0E E A A=A A=0.

2、This contradicts to the fact that A is invertible,so 0.1.Eigenvalue of Invertible Matrix1.Eigenvalue of Invertible MatrixLet be the eigenvector of matrix A corresponding to the eigenvalue,then =A1 A1 =1/So 1/is the eigenvalue of A-1,and matrix A-1 has the same eigenvector as matrix A.A A=Property 2.

3、If is the eigenvalue of A,then the eigenvalue of the following polynomial matrix (A)=a0E+a1A+amAmis ()=a0+a1+amm.Proof.Let be the eigenvector of A corresponding to the eigenvalue,thenA A=A(A )=A()A2 =2 2 2.Eigenvalue of Polynomial Matrix.Eigenvalue of Polynomial Matrix Similarly,k is the eigenvalue of Ak,and Ak has the same eigenvector as matrix A.(A)=(a0E+a1A+amAm)=(a0+a1+amm)Thus,()=a0+a1+amm is the eigenvalue of polynomial matrix(A)=a0E+a1A+amAm.Corollary.If is the eigenvalue of A,then the eigenvalue of the following polynomial matrix (A)=a-nA-n+a0E+a1A+amAmis ()=a-n-n+a-1-1+a0+a1+amm.