Abstract: When the extended matrix Rk(A) is not equal to its perturbation matrix R~k(A),the perturbation bounds under additvie perburtation and multiplicatvie perturbation are deducted,by means of singualr value decomposition,the decomposited subunitary facotrs of the extended matrix Rk(A),and its perturbation matrix R~k(A) in generalized polar decomposition under Frobenius norms.

Key words: extended matrix； generalized polar decomposition； Frobenius norm； subunitary matrix； singular value decomposition； perturbation bound；