Eigenvalues of 3x3 correlation matrix

Question

For a simple $2 \times 2$ correlation matrix, \begin{vmatrix}1&\rho\\\rho&1\\ \end{vmatrix} the eigenvalues are $1\pm\rho$. This is given a result in most textbooks.

Now, I am trying to find out the same for a $3 \times 3$ correlation matrix.\begin{vmatrix}1&\rho_1&\rho_2\\\rho_1&1&\rho_3\\\rho_2&\rho_3&1\\ \end{vmatrix}

I had a hard time factorizing the cubic characteristic equation. Thanks in advance for your help.

Answer 1

The characteristic polynomial is, up to sign, $$ \chi(t)=t^3 - 3t^2 + t( - r_1^2 - r_2^2 - r_3^2 + 3) + r_1^2 - 2r_1r_2r_3 + r_2^2 + r_3^2 - 1, $$ where I wrote $r_i=\rho_i$. The roots can be expressed with the usual cubic formula.