Show that the (m 1) by (m 1) tridiagonal method matrix A given by where λ > 0, is positive definite and diagonally dominant and

Show that the (m ˆ’ 1) by (m ˆ’ 1) tridiagonal method matrix A given by

where λ > 0, is positive definite and diagonally dominant and has eigenvalues

with corresponding eigenvectors v(i), where v(i)j = sin(ijπ/m).

-A, J=1-10j:1+1, 0, otherwise 4-1+4(sin )', for each i-l,1 for each 1, 2, . . . , m-1 , n-l,