char poly coefficients as power sums

Expanding det(xI - A) directly is expensive and clumsy. There is a cleaner route to the coefficients of the characteristic polynomial that uses only the traces of the powers A, A^2, A^3, and so on. The connecting machinery is Newton's identities, which relate the elementary symmetric functions of the eigenvalues (the char-poly coefficients, up to sign) to their power sums (the traces of powers).

Set p_j = tr(A^j) = sum lambda_i^j, the j-th power sum of the eigenvalues, and let e_j be the j-th elementary symmetric function, so chi(x) = x^n - e_1 x^(n-1) + e_2 x^(n-2) - ... up to (-1)^n e_n. Newton's identities give a recursion: e_1 = p_1, then k e_k = e_(k-1) p_1 - e_(k-2) p_2 + ... up to (-1)^(k-1) p_k. Knowing the traces p_1, ..., p_n hands you all the coefficients e_1, ..., e_n one at a time.

Crucially you compute tr(A^j) without ever finding the eigenvalues: just take matrix powers and sum diagonals. So the whole characteristic polynomial emerges from trace arithmetic, sidestepping both eigenvalue-finding and the permutation-sum expansion of the determinant. The Faddeev-LeVerrier algorithm packages exactly this idea into an elegant loop that also produces the adjugate and the inverse along the way.

Why it matters: it reveals that the char-poly coefficients are symmetric functions of the spectrum, it gives a determinant-free computational path well suited to symbolic computation, and it makes transparent identities like 'sum of eigenvalues = trace' and 'product of eigenvalues = determinant' as the first and last instances of the same family. Power sums and coefficients are two encodings of one spectrum.