n is an integer greater than 2 and φ = 2π/n. A is the n x n matrix (a_{ij}), where a_{ij} = cos(i + j)φ for i ≠ j, 1 + cos2jφ for i = j. Find det A.

**Solution**

A direct approach does not work (at least, I could not get it to work). However, if we let B be the n x n matrix (b_{ij}) with b_{ij} = cos(i + j)φ. Then the eigenvalues λ_{1}, λ_{2}, ... , λ_{n} of B satisfy the characteristic polynomial det(B - λ I) = 0. Hence det A = P(λ_{i} + 1).

For r = 1, 2, 3, ... , n, let v_{r} be the column vector (e^{iφr}, e^{2iφr}, e^{3iφr}, ... , e^{niφr}). Using cos sφ = (e^{isφ} + e^{-isφ})/2, we have that 2Bv_{r} is the column vector with jth element ( e^{iφ(j+1+r)} + e^{iφ(j+2+2r)} + e^{iφ(j+3+3r)} + ... + e^{iφ(j+n+nr)} ) + ( e^{iφ(-j-1+r)} + e^{iφ(-j-2+2r)} + e^{iφ(-j-3+3r)} + ... + e^{iφ(-j-n+nr)} ) = e^{iφj}(α + α^{2} + ... + α^{n}) + e^{-iφj}(β + β^{2} + ... + β^{n}), where α = e^{iφ(r+1)} and β = e^{iφ(r-1)}. If r is not 1 or n - 1, then α and β are nth roots of 1 other than 1 and hence α + α^{2} + ... + α^{n} = β + β^{2} + ... + β^{n} = 0, so 2Bv_{r} = 0, and so v_{r} is an eigenvector with eigenvalue 0. If r = 1, then β = 1, and so α + α^{2} + ... + α^{n} = 0, β + β^{2} + ... + β^{n} = n. Hence Bv_{1} = n/2 v_{n-1}. Similarly, if r = n - 1, then α = 1, and so α + α^{2} + ... + α^{n} = n, β + β^{2} + ... + β^{n} = 0. Hence Bv_{n-1} = n/2 v_{1}. So (v_{1} + v_{n-1})/2 [which is the column vector (cos φ, cos 2φ, cos 3φ, ... , cos nφ) ] is an eigenvector with eigenvalue n/2, and (v_{1} - v_{n-1})/2i [which is the column vector (sin φ, sin 2φ, ... , sin nφ) ] is an eigenvector with eigenvalue -n/2.

Finally, notice that the determinant with columns v_{r} is a Vandermonde determinant, whose value (using the well-known formula) is ∏_{r>s}(e^{irφ} - e^{isφ}) ≠ 0. So we have found n - 2 linearly independent eigenvectors for the eigenvalue 0, and hence 0 is an eigenvalue of multiplicity n - 2 (and there are no other eigenvalues).

Thus det A = ∏(λ_{i} + 1) = (1 + n/2)(1 - n/2) = -n^{2}/4 + 1.

© John Scholes

jscholes@kalva.demon.co.uk

14 Dec 1999