${\mathbf{U}} = \mathbf{C}^{1/2}$ and Its Invariants in Terms of $\mathbf{C}$ and Its Invariants

Abstract

We consider $N\times N$ tensors for $N= 3,4,5,6$. In the case $N=3$, it is desired to find the three principal invariants $i_{1}, i_{2}, i_{3}$ of ${\mathbf{U}}$ in terms of the three principal invariants $I_{1}, I_{2}, I_{3}$ of ${\mathbf{C}}={\mathbf{U}}^{2}$. Equations connecting the $i_{\alpha }$ and $I_{\alpha }$ are obtained by taking determinants of the factorisation

$$ \lambda ^{2}{\mathbf{I}}-{\mathbf{C}}= (\lambda {\mathbf{I}}-{\mathbf{U}})( \lambda {\mathbf{I}}+{\mathbf{U}}), $$

and comparing coefficients. On eliminating $i_{2}$ we obtain a quartic equation with coefficients depending solely on the $I_{\alpha }$ 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 ${\mathbf{U}}$ and ${\mathbf{U}}^{-1}$ are expressed solely in terms of ${\mathbf{C}}$, 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, ${\mathbf{U}}$ and ${\mathbf{U}}^{-1}$ are expressed in terms of powers of ${\mathbf{C}}$ 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 $i_{1}^{2}$. Sixteen of these roots are relevant, which we exhibit, but the other 16 are spurious. ${\mathbf{U}}$ and ${\mathbf{U}}^{-1}$ are expressed in terms of powers of ${\mathbf{C}}$. The cases $N>6$ are discussed.