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This is a natural generalization of a recent MSE question.

Let $X=(x_k)_{k\geq 1}$ be a sequence of real numbers, and $A=(a_{ij})_{i\geq 1,j\geq 1}$ be a real infinite matrix indexed by ${\mathbb N}_{>0}$. Suppose that "$AX=0$" ; I mean by that that for every $i\geq 1$, the series $\sum_{j=1}^{\infty}a_{ij}x_j$ is convergent with sum equal to zero.

Suppose also that for every $n\geq 1$, the upper left minor $A_n=(a_{ij})_{1\leq i \leq n,1\leq j \leq n }$ is invertible. Does it necessarily follow that all the $x_k$ are zero ?

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Ewan Delanoy
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1 Answers1

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No. An example is given by the following matrix: $$ A=\begin{pmatrix} 1 & -1 & 0 & 0 & \cdots \\ 0 & 1 & -1 & 0 & \cdots \\ 0 & 0 & 1 & -1 & \cdots \\ \vdots & \vdots & \vdots & \end{pmatrix} $$ with $X$ being the vector whose coordinates are all $1$. Then $AX=0$, and all upper-left square submatrices $A_n$ have determinant $1$.

Pierre-Guy Plamondon
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