Let h = 1/(N + 1). Define the tridiagonal matrix A = (aij) such that the...

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Let h = 1/(N + 1). Define the tridiagonal matrix A = (aij) such that the only nonzero entries are aii ai,i+1 = ai, i-1 Prove that the eigenpairs (k, uk) of A are = = 2 h2' -1 h2' 1 h2, 1 i N 1i N 1 2 i N 1 k N, Ak = 21/2/2 (1 - cos(kh)), uk = h2 Hint: you need to check that Auk = . sin(kh) sin(2kh) sin(3kh) : sin(Nkh) Let h = 1/(N + 1). Define the tridiagonal matrix A = (aij) such that the only nonzero entries are aii ai,i+1 = ai, i-1 Prove that the eigenpairs (k, uk) of A are = = 2 h2' -1 h2' 1 h2, 1 i N 1i N 1 2 i N 1 k N, Ak = 21/2/2 (1 - cos(kh)), uk = h2 Hint: you need to check that Auk = . sin(kh) sin(2kh) sin(3kh) : sin(Nkh)