Eigenvector in Non-Commutative Algebra

Abstract

$\newcommand{\Vector}[1]{\bar{#1}{}}$ $\newcommand{\Basis}[1]{\bar{\bar{#1}}{}}$ $\newcommand{\RC}{{}_*{}^*-}$ Let $\Basis e$ be a basis of vector space $V$ over non-commutative $D$-algebra $A$. Endomorhism $\Vector{\Basis eb}$ of vector space $V$ whose matrix with respect to given basis $\Basis e$ has form $Eb$ where $E$ is identity matrix is called similarity transformation with respect to the basis $\Basis e$. Let $V$ be a left $A$-vector space and $\Basis e$ be basis of left $A$-vector space $V$. The vector $v\in V$ is called eigenvector of the endomorphism \[\Vector f:V\rightarrow V\] with respect to the basis $\Basis e$, if there exists $b\in A$ such that \[ \Vector f\circ{v}= \Vector{\Basis eb} \circ{v} \] $A$-number $b$ is called eigenvalue of the endomorphism $\Vector f$ with respect to the basis $\Basis e$. There are two products of matrices: ${}_*{}^*$ (row column: $(ab)^i_j=a^i_kb^k_j$) and ${}^*{}_*$ (column row: $(ab)^i_j=a^k_jb^i_k$). $A$-number $b$ is called $\RC$ eigenvalue of the matrix $f$ if the matrix $f-bE$ is $\RC$ singular matrix. The $A$-number $b$ is called right $\RC$ eigenvalue if there exists the column vector $u$ which satisfies to the equality \[a{}_*{}^* u=ub\] The column vector $u$ is called eigencolumn for right $\RC$ eigenvalue $b$. The $A$-number $b$ is called left $\RC$ eigenvalue if there exists the row vector $u$ which satisfies to the equality \[u{}_*{}^* a=bu\] The row vector $u$ is called eigenrow for right $\RC$ eigenvalue $b$. The set $\RC$ $\mathrm{spec}(a)$ of all left and right $\RC$ eigenvalues is called $\RC$ spectrum of the matrix a.