= Consider the set of orthogonal polynomials with following recurrence relations fo(x) = 1, (n+1) fn+1(x)...
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= Consider the set of orthogonal polynomials with following recurrence relations fo(x) = 1, (n+1) fn+1(x) = (2n+ 1)xƒn(x) – nfn-1(x). The polynomials are orthogonal with respect to the inner product 1₁ fn(x) fm (x) dx 2n²18nm. Restrict the polynomials to n=0,2,4 and construct the 3×3 matrix for the operator A = x²d²/dx² - xd/dx in this basis of fo, f2, f4. Consider the following matrix - ( A = 0 a²k 0 ak 0 ik 0-ik 0 (5) Find the two values of a for which the matrix A is Hermitian. Find the eigenvalues and eigenvectors of this matrix for both these values. = Consider the set of orthogonal polynomials with following recurrence relations fo(x) = 1, (n+1) fn+1(x) = (2n+ 1)xƒn(x) – nfn-1(x). The polynomials are orthogonal with respect to the inner product 1₁ fn(x) fm (x) dx 2n²18nm. Restrict the polynomials to n=0,2,4 and construct the 3×3 matrix for the operator A = x²d²/dx² - xd/dx in this basis of fo, f2, f4. Consider the following matrix - ( A = 0 a²k 0 ak 0 ik 0-ik 0 (5) Find the two values of a for which the matrix A is Hermitian. Find the eigenvalues and eigenvectors of this matrix for both these values.
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