Let us consider the matrix $\mathbf{A}$ of dimension $n$. The eigenvalues of $\mathbf{A}$ are defined through the matrix equation $$\mathbf{A}\mathbf{x}^{(\nu)} = \lambda^{(\nu)}\mathbf{x}^{(\nu)},$$ where $\lambda^{(\nu)}$ are the eigenvalues and $\mathbf{x}^{(\nu)}$ the corresponding eigenvectors. Unless otherwise stated, when we use the wording eigenvector we mean the right eigenvector. The left eigenvalue problem is defined as $$\mathbf{x}^{(\nu)}_L\mathbf{A} = \lambda^{(\nu)}\mathbf{x}^{(\nu)}_L$$ The above right eigenvector problem is equivalent to a set of $n$ equations with $n$ unknowns $x_i$.