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Use 伴随/自伴 for "adjoint" with footnote distinguishing from adjugate #67

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jeanwsr merged 2 commits into from
Feb 26, 2026
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Use 伴随/自伴 for "adjoint" with footnote distinguishing from adjugate #67

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Copilot Feb 26, 2026
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Revert adjoint to 伴随/自伴, add footnote distinguishing from adjugate
Copilot Feb 26, 2026
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14 changes: 7 additions & 7 deletions Chaps/Chap1.tex
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Expand Up @@ -296,7 +296,7 @@ \subsection{矩阵}
\end{equation}

下面我们将介绍一些重要的定义。
一个$N\times M$矩阵 $\mbf A$的\emph{转置共轭}(adjoint)，
一个$N\times M$矩阵 $\mbf A$的\emph{伴随}(adjoint)\footnote{本书中 adjoint 译为"伴随"，与量子力学中的惯例一致，指矩阵的转置共轭（即各元素取复共轭后再转置）。注意这与线性代数中的"伴随矩阵"（adjugate，即由代数余子式构成的矩阵）含义不同。}，
记为$\mbf{A}^\dagger$，是一个$M \times N$的矩阵，其元素为：
\begin{equation}
\left(\mbf{A}^\dagger\right)_{ij} = A_{ji}^{\ast}
Expand All @@ -305,13 +305,13 @@ \subsection{矩阵}
即对矩阵$\mbf A$的每个元素取复共轭，
并交换矩阵的行和列。
如果$\mbf A$为实矩阵，
则其转置共轭称为$\mbf A$的\emph{转置}。
则其伴随称为$\mbf A$的\emph{转置}。

\exercise{
若$\mbf A$为维度$N \times M$的矩阵，$\mbf B$为$M \times K$矩阵，证明$\left( \mbf{A} \mbf{B} \right)^{\dagger} = \mbf{B}^\dagger \mbf{A}^\dagger$。
}

一个列矩阵的转置共轭为一个\emph{行矩阵}，
一个列矩阵的伴随为一个\emph{行矩阵}，
其元素为原矩阵元素的复共轭：
\begin{equation}
{\mbf a}^{\dagger} =
Expand All @@ -335,7 +335,7 @@ \subsection{矩阵}
通过与\autoref{eq:1.4}相比，
我们注意到，如果$\mbf a$和$\mbf b$均为实矩阵且$M$值为3时，
上式即为两个三维矢量的标量积。
对\autoref{eq:1.24}两边分别取其转置共轭，
对\autoref{eq:1.24}两边分别取其伴随，
并引用\autoref{ex:1.3} 的结论，
我们得到如下关系：
\begin{equation}
Expand Down Expand Up @@ -381,14 +381,14 @@ \subsection{矩阵}
\label{eq:1.35}
\end{equation}

\item 若矩阵$\mbf A$的逆矩阵与其转置共轭相等，则$\mbf A$为\emph{幺正}矩阵：
\item 若矩阵$\mbf A$的逆矩阵与其伴随相等，则$\mbf A$为\emph{幺正}矩阵：
\begin{equation}
{\mbf A}^{-1} = \madj{A}
\label{eq:1.36}
\end{equation}
一个实幺正矩阵被称为\emph{正交}阵。

\item 是自身的转置共轭(self-adjoint)的矩阵称为\emph{Hermitian}矩阵，即：
\item 自伴(self-adjoint)矩阵称为\emph{Hermitian}矩阵，即：
\begin{subequations}
\begin{equation}
\madj{A} = {\mbf A} \\
Expand Down Expand Up @@ -561,7 +561,7 @@ \subsection{N 维复矢量空间}
\end{equation}
这里 $\mbf a$是抽象的矢量 $\ket{a}$在基组 $\left\{\ket{i}\right\}$ 中的矩阵表示。
\autoref{eq:1.27}提到，
一个列矩阵$\mbf a$的转置共轭$\madj{a}$是一个行矩阵：
一个列矩阵$\mbf a$的伴随$\madj{a}$是一个行矩阵：
\begin{equation}
\madj{a} = \begin{pmatrix}
a_1^\ast & a_2^\ast &\cdots &a_N^\ast
Expand Down