1.5.l Eigenvalue and Eigenvector(4) If A is nonsingular, with eigenvalues a, (i=l,2, .-,n) ,1are the eigenvalues of A-1then元(5)If the characteristic polynomial of A can be written infactored form as.n[21 -A=(-)~I(-)i=α+1of thethe parameter α is called the algebraic multiplicityeigenvalue 2 -
1.5.1 Eigenvalue and Eigenvector
1.5.l Eigenvalue and Eigenvector(6) If n-rank(2I - A)= β, the parameter β is called thegeometricalmultiplicity of the eigenvalue 2 .Any nonzero vector V, which satisfies the matrix equation(2,I-A)V, = 0 of A associated with eigenvalueis called theeigenvector元, (i=1,2,.,n). If A has distinct eigenvalues, the eigenvectors can besolved directly by the equation above.It should be pointed out that if A has multiplicityeigenvalues, not all eigenvectors can be found
1.5.1 Eigenvalue and Eigenvector
1.5.l Eigenvalue and EigenvectorAny nonzero vector V, which satisfies the matrix equation(a,I - A)V, = 0eigenvectoris called the r of A associated with eigenvalue, (i=1,2,..-,n). +Let us assume that is the m multiplicity eigenvalue ofA and the remainingdistinct eigenvaluesaren-mm+1,2Based on 入m+1, 2m+2a,, we can find n-m linearlyVVindependenteigenvectorsV171-13m+2
1.5.1 Eigenvalue and Eigenvector