论文标题
非交通代数的特征向量
Eigenvector in Non-Commutative Algebra
论文作者
论文摘要
$ \ newCommand {\ vector} [1] {\ bar {＃1} {}}} $ $ \ $ \ newCommand {\ baseban} [1] {\ bar {\ bar {＃1}}}}} {}} {}}}} $ $ \ $ \ $ \ newcommand {\ rc} $} $} $} $} E $是矢量空间$ v $的基础，超过了非交通$ d $ -Algebra $ a $。 vector space $ v $的endorhism $ \ vector {\ basite eb} $，其矩阵相对于给定的基础$ \ basite e $具有$ eb $，其中$ e $ as $ e $ as sidentity矩阵称为相似性转换相对于基础$ \基础e $。让$ v $为左$ a $ a-vector Space，而$ \ base e $是左$ a $ a-vector space $ v $的基础。 v $中的向量$ v \在基于$ \基础e $的情况下称为内型\ [\ vector f：v \ vector f：v \ rightArrow v \]，如果存在$ b \ in $ b \ in $ sOS as as as as os as as os sO sOS as as as sOS as as as as as as as as as as as as as as as as as as as as as as as as as as as as as as as SOS \ [\ Vector F \ Vector F \ Vection F \ circ {vection eb basis baseb cock $ cy \ $ c $ cy \ $ cy}相对于基础$ \基础e $，被称为内态$ \ vector f $的特征值。矩阵有两种产品：$ {} _*{}^*$（行列：$（ab）^i_j = a^i_kb^k_j $）和$ {}^*{}^*{} {} _*$（列行：$（ab）^i_j = a^k_jb^i_k^i_k $）。如果矩阵$ f-be $是$ \ rc $ singular matrix，则$ a $ -number $ b $称为矩阵$ f $的$ \ rc $ eigenvalue。 $ a $ -number $ b $称为正确的$ \ rc $ eigenvalue如果存在列vector $ u $，它满足了equality \ [a {a {} _* {} _* {}^* u = ub = ub \ ub = ub $ $ $ u $，称为eigencolumn of eigencolumn of eigencolumn，for rc $ \ rc $ eigenvalue $ b $。 $ a $ -number $ b $称为左$ \ rc $ eigenvalue，如果存在的行vector $ u $满足equality \ [u {} _* {} _* {} _ = a = a = a = a = a = bu \ \ \ \ rc $ u $的eigenrow for Right $ \ rc $ eigenvalue $ eigenvalue $ b $。所有左右$ \ rc $ eigenvalues的套装$ \ rc $ $ \ $ \ mathrm {spec}（a）$称为矩阵a的$ \ rc $ spectrum。
$\newcommand{\Vector}[1]{\bar{#1}{}}$ $\newcommand{\Basis}[1]{\bar{\bar{#1}}{}}$ $\newcommand{\RC}{{}_*{}^*-}$ 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.
