U = C1/2 and Its Invariants in Terms of C and Its Invariants

We consider N x N tensors for N = 3, 4, 5, 6. In the case N = 3, it is desired to find the three principal invariants i1, i2, i3 of U in terms of the three principal invariants I-1, I-2, I-3 of C = U-2. Equations connecting the ia and Ia are obtained by taking determinants of the factorisation lambda I-2- C = (lambda I- U)(lambda I+ U), and comparing coefficients. On eliminating i(2) we obtain a quartic equation with coefficients depending solely on the Ia whose largest root is i(1). Similarly, we may obtain a quartic equation whose largest root is i(2). For N = 4 we find that i(2) is once again the largest root of a quartic equation and so all the i(alpha) are expressed in terms of the I-alpha. Then U and U-1 are expressed solely in terms of C, as for N = 3. For N = 5 we find, but do not exhibit, a twentieth degree polynomial of which i(1) is the largest root and which has four spurious zeros. We are unable to express the i(alpha) in terms of the I-alpha for N = 5. Nevertheless, U and U-1 are expressed in terms of powers of C with coefficients now depending on the i(alpha). For N = 6 we find, but do not exhibit, a 32 degree polynomial which has largest root i(2)(1). Sixteen of these roots are relevant, which we exhibit, but the other 16 are spurious. U and U-1 are expressed in terms of powers of C. The casesN > 6 are discussed.