While working on a problem in Mechanics I came across this closed form solution for the principal directions (which are the eigenvectors of an Hermitian matrix), which I can't wrap my head around. Beware! (index notation).
Let $\sigma$ be an eigenvalue of the stress tensor $\sigma_{ij}$ (known to be Hermitian), a non-trivial solution for the principal direction $n_j$ associated with $\sigma$ must satisfy
\begin{equation} (\sigma_{ij} - \sigma\delta_{ij}) n_j = 0 \label{eq:condition1} \tag{1} \end{equation} \begin{equation} n_jn_j =1 \label{eq:condition2} \tag{2} \end{equation}
This is the part where I can't follow
Let \begin{equation} K = n_1/C_1 = n_2/C_2 = n_3/C_3 \label{eq:K} \tag{3} \end{equation}
(how does one arrive at such equality?)
Where $C_i$ is the cofactor of the $i$th column of $\sigma_{ij} - \sigma\delta_{ij}$.
From the first equation of the system given by $\eqref{eq:condition1}$ we have \begin{equation} (\sigma_{11} - \sigma)C_1 K + C_2 K \sigma_{12} + C_3 K\sigma_{13}=0 \label{eq:1st} \tag{4} \end{equation} Since the determinant of $\sigma_{ij} - \sigma\delta_{ij}$ must vanish, expressed in terms of cofactors: \begin{equation} (\sigma_{11} - \sigma)C_1 + C_2 \sigma_{12} + C_3 \sigma_{13}=0 \label{eq:determinant} \tag{5} \end{equation}
Which is exactly $\eqref{eq:1st}$, indicating the validity of $\eqref{eq:K}$.
$K$ is obtained by subtituting in $\eqref{eq:condition2}$: \begin{equation} K = \frac{1}{\sqrt{C_iC_i}} \end{equation} Thus the eigenvector is given then by \begin{equation} n_j = \frac{C_j}{\sqrt{C_iC_i}} \end{equation}
My question refers to the formulation of $\eqref{eq:K}$, how do we propose such a factor? I understand that we can work up from $\eqref{eq:determinant}$ up to $\eqref{eq:1st}$ multiplying by an arbitrary constant, but I can't still wrap my head around $\eqref{eq:K}$.
Also this approach fails suddenly when the cofactor vanishes. No more information is given by the author, the bibliography where this supposedly appears is
Durelli, A J, Philips, E A y Tsao, C H, Introduction to the Theoretical and Experimental Analysis of Stress and Strain, McGraw-Hill, New York, 1958
Sadly I can't get my hands on a copy.