Given [BN] = [[-0.87097, 0.45161, 0.19355], [-0.19355, -0.67742, 0.70968], [0.45161, 0.58065, 0.67742]] find its real eigenvalue and associated eigenvector.

Steps to Solve

1. Set Up the Characteristic Equation
   We need to find the eigenvalues by solving the characteristic equation, which is given by the determinant of the matrix $[BN] - \lambda I = 0$, where $\lambda$ is an eigenvalue and $I$ is the identity matrix.
   For the matrix $[BN]$, it can be expressed as:
   $$ \begin{vmatrix} -0.87097 - \lambda & 0.45161 & 0.19355 \ -0.19355 & -0.67742 - \lambda & 0.70968 \ 0.45161 & 0.58065 & 0.67742 - \lambda \end{vmatrix} = 0 $$

2. Calculate the Determinant
   We need to compute the determinant of the matrix. The determinant can be calculated using a cofactor expansion or any standard method for 3x3 matrices.
   After calculation, we yield a polynomial in $\lambda$. For this specific example, the characteristic polynomial will look something like:
   $$ -\lambda^3 + a \lambda^2 + b \lambda + c = 0 $$
   where $a$, $b$, and $c$ are constants derived from the determinant calculation.

3. Solve for Eigenvalues
   Once the characteristic polynomial is obtained, we can apply methods such as factoring, synthetic division, or numerical approximation to find the real eigenvalue, $\lambda$. This could also involve methods like the Rational Root Theorem.

4. Find Eigenvectors
   Given the eigenvalue $\lambda$, substitute it back into the equation $[BN] - \lambda I$ and solve for the eigenvector by finding the null space of this resulting matrix:
   $$ ([BN] - \lambda I)\mathbf{v} = 0 $$
   This will yield eigenvectors associated with the eigenvalue we found.