Method to calculate solution of a linear equation system?

Question

I am searching a solution method for the following equation system of equation systems:

Let $A, B \in \mathbb{R}^{n \times n}$ be s.p.d. Matrices and $O$ be the zero matrix of the same size. Further let $f\in\mathbb{R}^n$ be given vector and $0$ the zero vector. I am searching for a numerical method to find the solution vectors $x_1, x_2 \in\mathbb{R}^n$ such that

\begin{equation} \begin{pmatrix} A & B \\ O & A \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} f \\ 0 \end{pmatrix} \end{equation}

I tried to solve this in Python with (scipy.linalgscipy.linalg: lu_factor, lu_solve). The problem, I only get the trivial solution $x_2 = 0$. Is there a way to solve the equation system of equation systems for non-trivial solutions?

Answer 1

There is no way because there is no solution to the given system with $x_2 \neq 0$. This is because the second block of the equation system reads $Ax_2 = 0$, which has no non-trivial solutions because $A$ is SPD (Otherwise $x_2$ would be an eigenvector of $A$ corresponding to a zero eigenvalue).