adjugate
MatrixAdjoint
2002-11-17 18:09:27
2006-09-07 22:39:28
Definition
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\newcommand{\adj}{\operatorname{adj}}
\newcommand{\adjA}{\adj(A)}
The \emph{adjugate}, $\adjA$, of an $n\times n$
matrix $A$, is the $n\times n$ matrix
\begin{equation}
\label{eq:def1}
\adjA_{ij} = (-1)^{i+j}\, M_{\!ji}(A)
\end{equation}
where $M_{\!ji}(A)$ is the indicated minor of $A$ (the determinant
obtained by deleting row $j$ and column $i$ from $A$). The adjugate
is also known as the \emph{classical adjoint}, to distinguish it from
the \PMlinkname{usual usage of ``adjoint''}{AdjointEndomorphism} which
denotes the conjugate transpose operation.
An equivalent characterization of the adjugate is the following:
\begin{equation}
\label{eq:def2}
\adjA A = \det(A) I.
\end{equation}
The equivalence of \eqref{eq:def1} and \eqref{eq:def2} follows easily
from the \PMlinkname{multi-linearity
properties}{DeterminantAsAMultilinearMapping} of the determinant.
Thus, the adjugate operation is closely related to the matrix inverse.
Indeed, if $A$ is invertible, the adjugate can be defined as
\[ \adjA = \det(A)A^{-1} \]
Yet another definition of the adjugate is the following:
\begin{align}
\label{eq:def3}
\adjA = p_{n-1}(A) I &- p_{n-2}(A) A + p_{n-3}(A) A^2 - \cdots \\ \nonumber
& + (-1)^{n-2}p_1(A) A^{n-2} + (-1)^{n-1}A^{n-1},
\end{align}
where $p_1(A)=\operatorname{tr}(A), p_2(A),\ldots, p_n(A) = \det(A)$
are the elementary invariant polynomials of
$A$. The latter arise as
coefficients in the
characteristic polynomial $p(t)$ of $A$, namely
\[p(t) = \det(t I - A) = t^n - p_1(A) t^{n-1} + \cdots + (-1)^n p_n(A).\]
The equivalence of \eqref{eq:def2} and \eqref{eq:def3} follows from
the Cayley-Hamilton theorem. The latter states that $p(A)=0$, which
in turn implies that
\[A ( A^{n-1} - p_1(A) A^{n-2} + \cdots + (-1)^{n-1} p_{n-1}(A) ) =
(-1)^{n-1} \det(A) I\]
The adjugate operation enjoys a number of notable
properties:
\begin{align}
&\adj(AB) =\adj(B)\adj(A),\\
&\adj(A^t) = \adjA^t,\\
&\det(\adjA) = \det(A)^{n-1}.
\end{align}