| Version 13 |
Version 12 |
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The \emph{adjugate}, $\adjA$, of an $n\times n$
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The adjugate, $\adjA$, of an $n\times n$
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| matrix $A$, is the $n\times n$ matrix |
matrix $A$, is the $n\times n$ matrix |
| \begin{equation} |
\begin{equation} |
| \label{eq:def1} |
\label{eq:def1} |
| \adjA_{ij} = (-1)^{i+j}\, M_{\!ji}(A) |
\adjA_{ij} = (-1)^{i+j}\, M_{\!ji}(A) |
| \end{equation} |
\end{equation} |
| where $M_{\!ji}(A)$ is the indicated minor of $A$ (the determinant |
where $M_{\!ji}(A)$ is the indicated minor of $A$ (the determinant |
| obtained by deleting row $j$ and column $i$ from $A$). The adjugate |
obtained by deleting row $j$ and column $i$ from $A$). The adjugate |
| is also known as the \emph{classical adjoint}, to distinguish it from |
is also known as the \emph{classical adjoint}, to distinguish it from |
| the \PMlinkname{usual usage of ``adjoint''}{AdjointEndomorphism} which |
the \PMlinkname{usual usage of ``adjoint''}{AdjointEndomorphism} which |
| denotes the conjugate transpose operation. |
denotes the conjugate transpose operation. |
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| An equivalent characterization of the adjugate is the following: |
An equivalent characterization of the adjugate is the following: |
| \begin{equation} |
\begin{equation} |
| \label{eq:def2} |
\label{eq:def2} |
| \adjA A = \det(A) I. |
\adjA A = \det(A) I. |
| \end{equation} |
\end{equation} |
| The equivalence of \eqref{eq:def1} and \eqref{eq:def2} follows easily |
The equivalence of \eqref{eq:def1} and \eqref{eq:def2} follows easily |
| from the \PMlinkname{multi-linearity |
from the \PMlinkname{multi-linearity |
| properties}{DeterminantAsAMultilinearMapping} of the determinant. |
properties}{DeterminantAsAMultilinearMapping} of the determinant. |
| Thus, the adjugate operation is closely related to the matrix inverse. |
Thus, the adjugate operation is closely related to the matrix inverse. |
| Indeed, if $A$ is invertible, the adjugate can be defined as |
Indeed, if $A$ is invertible, the adjugate can be defined as |
| \[ \adjA = \det(A)A^{-1} \] |
\[ \adjA = \det(A)A^{-1} \] |
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| Yet another definition of the adjugate is the following: |
Yet another definition of the adjugate is the following: |
| \begin{align} |
\begin{align} |
| \label{eq:def3} |
\label{eq:def3} |
| \adjA = p_{n-1}(A) I &- p_{n-2}(A) A + p_{n-3}(A) A^2 - \dots \\ \nonumber |
\adjA = p_{n-1}(A) I &- p_{n-2}(A) A + p_{n-3}(A) A^2 - \dots \\ \nonumber |
| & + (-1)^{n-2}p_1(A) A^{n-2} + (-1)^{n-1}A^{n-1}, |
& + (-1)^{n-2}p_1(A) A^{n-2} + (-1)^{n-1}A^{n-1}, |
| \end{align} |
\end{align} |
| where $p_1(A)=\operatorname{tr}(A), p_2(A),\ldots, p_n(A) = \det(A)$ |
where $p_1(A)=\operatorname{tr}(A), p_2(A),\ldots, p_n(A) = \det(A)$ |
| are the elementary invariant polynomials of |
are the elementary invariant polynomials of |
| $A$. The latter arise as |
$A$. The latter arise as |
| coefficients in the |
coefficients in the |
| characteristic polynomial $p(t)$ of $A$, namely |
characteristic polynomial $p(t)$ of $A$, namely |
| \[p(t) = \det(t I - A) = t^n - p_1(A) t^{n-1} + ... + (-1)^n p_n(A).\] |
\[p(t) = \det(t I - A) = t^n - p_1(A) t^{n-1} + ... + (-1)^n p_n(A).\] |
| The equivalence of \eqref{eq:def2} and \eqref{eq:def3} follows from |
The equivalence of \eqref{eq:def2} and \eqref{eq:def3} follows from |
| the Cayley-Hamilton theorem. The latter states that $p(A)=0$, which |
the Cayley-Hamilton theorem. The latter states that $p(A)=0$, which |
| in turn implies that |
in turn implies that |
| \[A ( A^{n-1} - p_1(A) A^{n-2} + ... + (-1)^{n-1} p_{n-1}(A) ) = |
\[A ( A^{n-1} - p_1(A) A^{n-2} + ... + (-1)^{n-1} p_{n-1}(A) ) = |
| (-1)^{n-1} \det(A) I\] |
(-1)^{n-1} \det(A) I\] |
|
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| The adjugate operation enjoys a number of notable |
The adjugate operation enjoys a number of notable |
| properties: |
properties: |
| \begin{align} |
\begin{align} |
| &\adj(AB) =\adj(B)\adj(A),\\ |
&\adj(AB) =\adj(B)\adj(A),\\ |
| &\adj(A^t) = \adjA^t,\\ |
&\adj(A^t) = \adjA^t,\\ |
| &\det(\adjA) = \det(A)^{n-1}. |
&\det(\adjA) = \det(A)^{n-1}. |
| \end{align} |
\end{align} |
