1 Prove that if A is diagonally dominant and if Q is chosen

1. Prove that if A is diagonally dominant and if Q is chosen as in the Jacobi method, then rho(I - Q^-1A)

Solution

If A is diagonally dominant, then A is non singular. Also the diagonal elements are the maximum.

Q is the matrix selected from largest elements and forming the diagonals of Q. Obviously Q has rank as that of A

Q will be a diagonal matrix with elements of A as diagonals the same elements.

Obviously Q inverse also will be a diagonal matrix containing the reciprocals of Diagonal elements of A.

Q inverse A cannot be equal to 1 as A has some more elements than in Q

Hence Q inverse A >1

Or I -Q inverse A will have rank <1.