For a square matrix \(A\), an eigenvector \(v\) is a non-zero vector that, when multiplied by \(A\), yields a scaled version of \(v\). The scaling factor is the eigenvalue \(\lambda\).

This relationship is defined by the equation: \(Av = \lambda v\).

To find the eigenvalues:

1. Rearrange the equation to \((A - \lambda I)v = 0\), where \(I\) is the identity matrix.

2. Since \(v\) is non-zero, the matrix \((A - \lambda I)\) must be singular, meaning its determinant is zero.

3. Solve the characteristic equation, \(\det(A - \lambda I) = 0\), for \(\lambda\). The solutions are the eigenvalues.

To find the eigenvectors:

For each eigenvalue \(\lambda\), substitute it back into the equation \((A - \lambda I)v = 0\) and solve for the vector \(v\). This is equivalent to finding the null space of the matrix \((A - \lambda I)\).