| Version 4 |
Version 3 |
| The \emph{adjoint} or \emph{classical adjoint}\footnote{this term is to distinguish this sense from the conjugate transpose over the complexes sense, which is more recent and explored \PMlinkname{here}{AdjointEndomorphism}.} $A^*$ of a square matrix $A$ is given by |
The \emph{adjoint} or \emph{classical adjoint}\footnote{this term is to distinguish this sense from the conjugate transpose over the complexes sense, which is more recent and explored \PMlinkname{here}{AdjointEndomorphism}.} $M^*$ of a square matrix $M$ is given by |
| $$ A^*_{ij} = \operatorname{cof}_{ji}(A) $$ |
$$ M^*_{ij} = \operatorname{cof}_{ij}(M) $$ |
| where $\operatorname{cof}_{ji}(A)$ denotes the $j, i$th cofactor of $A$. |
where $\operatorname{cof}_{ij}(M)$ denotes the $i, j$th cofactor of $M$. Note that sometimes the transpose of the right-hand side is taken and included in the definition of the adjoint. |
| The adjoint is closely related to the matrix inverse, as |
The adjoint is closely related to the matrix inverse. |
| $$ A A^* = \det(A) I $$ |
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| characterizes $A^*$ for $A$ invertible. |
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| \subsection{Property} |
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| Let |
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| $$p(t) = \det(t I - A) = t^n - p_1(A) t^{n-1} + ... + (-1)^n \det(A)$$ |
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| be the characteristic polynomial of $A$, where $p_1(A), p_2(A),\ldots p_n(A) = \det(A)$ are the fundamental invariant polynomials of $A$\footnote{Note that $p_1(A) = \operatorname{tr}(A)$, the trace of $A$.}. |
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| From $p(A) = 0$ we get that |
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| $$A ( A^{n-1} - p_1(A) A^{n-2} + ... + (-1)^{n-1} p_{n-1}(A) ) = (-1)^{n-1} \det(A) I$$ |
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| so we have |
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| $$A^* = p_{n-1}(A) I - p_{n-2}(A) A + p_{n-3}(A) A^2 - \dots + (-1)^{n-1}p_1(A) A^{n-1}$$ |
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