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?

  • $\begingroup$ That looks like a saddle point system / Lagrange multiplier system gone awry $[\mathbf A \mathbf B; \mathbf B^T \mathbf 0]$. I'd double check your reference material / formulation, then ask a followup. There are special methods (both iterative and direct) for those kinds of systems. $\endgroup$ – rchilton1980 Nov 6 at 20:34

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).

  • $\begingroup$ Oh you are totally correct. I overlooked the fact of spd. Maybe something in my calculation is not correct. I will write again! $\endgroup$ – Kerem Nov 6 at 11:40

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