- Ax Ax For An Invertible Matrix A Prove That A And A 1 Have The Same Eigenvectors How Are The Eigenvalues Of A 1 (106.29 KiB) Viewed 31 times
Ax = Ax = = = For an invertible matrix A, prove that A and A-1 have the same eigenvectors. How are the eigenvalues of A related to the eger Letting x be an eigenvector of A gives Ax = ix for a corresponding eigenvalue 2. Using matrix operations and the properties of Ax = 2x ax az Ax = 2x AxA-1 = IxA-1 A-1 Ax = A-12x A/(Ax) A/(x) AX/A = x A 2A-2 IX = 2A-1x (A/A)X = (A/2) A XI = 2A-1X Ix = (A/2) x = 1A-1x Ix = x2-1 x = 2A-1X = λΑ και x = x2-1 A-2x = 1x A-1x = 1x ** A-2x = A-2x = 2x ОХАА-1 o ) (A/A)X = ix--- = = = x = * 美 This shows that -Select-- v is an eigenvector of A-1 with eigenvalue -Select-