Home > ELA > Vol. 32 (2017)
Keywords
common invariant subspaces, common eigenvectors, quantifier elimination, effective Nullstellensatz, quantum information theory
Abstract
This article presents a computable criterion for the existence of a common invariant subspace of $n\times n$ complex matrices $A_{1}, \dots ,A_{s}$ of a fixed dimension $1\leq d\leq n$. The approach taken in the paper is model-theoretic. Namely, the criterion is based on a constructive proof of the renowned Tarski's theorem on quantifier elimination in the theory $\ACF$ of algebraically closed fields. This means that for an arbitrary formula $\varphi$ of the language of fields, a quantifier-free formula $\varphi'$ such that $\varphi\lra\varphi'$ in $\ACF$ is given explicitly. The construction of $\varphi'$ is elementary and based on the effective Nullstellensatz. The existence of a common invariant subspace of $A_{1},\dots,A_{s}$ of dimension $d$ can be expressed in the first-order language of fields, and hence, the constructive version of Tarski's theorem yields the criterion. In addition, some applications of this criterion in quantum information theory are discussed.
Recommended Citation
Pastuszak, Grzegorz.
(2017),
"The common invariant subspace problem and Tarski’s theorem",
Electronic Journal of Linear Algebra,
Volume 32, pp. 343-356.
DOI: https://doi.org/10.13001/1081-3810.3439
Abstract