Adjugate matrix

Inlinear algebra,theadjugateof asquare matrix $A$ is thetransposeof itscofactor matrixand is denoted by $adj(A)$ .^[1]^[2]It is also occasionally known asadjunct matrix,^[3]^[4]or "adjoint",^[5]though the latter term today normally refers to a different concept, theadjoint operatorwhich for a matrix is theconjugate transpose.

The product of a matrix with its adjugate gives adiagonal matrix(entries not on the main diagonal are zero) whose diagonal entries are thedeterminantof the original matrix:

\mathbf {A} \operatorname {adj} (\mathbf {A} )=\det(\mathbf {A} )\mathbf {I},

where $I$ is theidentity matrixof the same size as $A$ .Consequently, the multiplicative inverse of aninvertible matrixcan be found by dividing its adjugate by its determinant.

Definition[edit]

Theadjugateof $A$ is thetransposeof thecofactor matrix $C$ of $A$ ,

\operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}.

In more detail, suppose $R$ is a unitalcommutative ringand $A$ is an $n \times n$ matrix with entries from $R$ .The $(i, j)$ -minorof $A$ ,denoted $M ij$ ,is thedeterminantof the $(n - 1) \times (n - 1)$ matrix that results from deleting row $i$ and column $j$ of $A$ .Thecofactor matrixof $A$ is the $n \times n$ matrix $C$ whose $(i, j)$ entry is the $(i, j)$ cofactorof $A$ ,which is the $(i, j)$ -minor times a sign factor:

\mathbf {C} =\left((-1)^{i+j}\mathbf {M} _{ij}\right)_{1\leq i,j\leq n}.

The adjugate of $A$ is the transpose of $C$ ,that is, the $n \times n$ matrix whose $(i, j)$ entry is the $(j, i)$ cofactor of $A$ ,

\operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}=\left((-1)^{i+j}\mathbf {M} _{ji}\right)_{1\leq i,j\leq n}.

Important consequence[edit]

The adjugate is defined so that the product of $A$ with its adjugate yields adiagonal matrixwhose diagonal entries are the determinant $det(A)$ .That is,

\mathbf {A} \operatorname {adj} (\mathbf {A} )=\operatorname {adj} (\mathbf {A} )\mathbf {A} =\det(\mathbf {A} )\mathbf {I},

where $I$ is the $n \times n$ identity matrix.This is a consequence of theLaplace expansionof the determinant.

The above formula implies one of the fundamental results in matrix algebra, that $A$ isinvertible if and only if $det(A)$ is aninvertible elementof $R$ .When this holds, the equation above yields

{\begin{aligned}\operatorname {adj} (\mathbf {A} )&=\det(\mathbf {A} )\mathbf {A} ^{-1},\\\mathbf {A} ^{-1}&=\det(\mathbf {A} )^{-1}\operatorname {adj} (\mathbf {A} ).\end{aligned}}

Examples[edit]

1 × 1 generic matrix[edit]

Since the determinant of a 0 × 0 matrix is 1, the adjugate of any 1 × 1 matrix (complexscalar) is $\mathbf {I} ={\begin{bmatrix}1\end{bmatrix}}$ .Observe that $\mathbf {A} \operatorname {adj} (\mathbf {A} )=\mathbf {A} \mathbf {I} =(\det \mathbf {A} )\mathbf {I}.$

2 × 2 generic matrix[edit]

The adjugate of the 2 × 2 matrix

\mathbf {A} ={\begin{bmatrix}a&b\\c&d\end{bmatrix}}

is

\operatorname {adj} (\mathbf {A} )={\begin{bmatrix}d&-b\\-c&a\end{bmatrix}}.

By direct computation,

\mathbf {A} \operatorname {adj} (\mathbf {A} )={\begin{bmatrix}ad-bc&0\\0&ad-bc\end{bmatrix}}=(\det \mathbf {A} )\mathbf {I}.

In this case, it is also true that $det$ ( $adj$ (A)) = $det$ (A) and hence that $adj$ ( $adj$ (A)) =A.

3 × 3 generic matrix[edit]

Consider a 3 × 3 matrix

\mathbf {A} ={\begin{bmatrix}a_{1}&a_{2}&a_{3}\\b_{1}&b_{2}&b_{3}\\c_{1}&c_{2}&c_{3}\end{bmatrix}}.

Its cofactor matrix is

\mathbf {C} ={\begin{bmatrix}+{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}\\\\-{\begin{vmatrix}a_{2}&a_{3}\\c_{2}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{3}\\c_{1}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{2}\\c_{1}&c_{2}\end{vmatrix}}\\\\+{\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{3}\\b_{1}&b_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\end{bmatrix}},

where

{\begin{vmatrix}a&b\\c&d\end{vmatrix}}=\det \!{\begin{bmatrix}a&b\\c&d\end{bmatrix}}.

Its adjugate is the transpose of its cofactor matrix,

\operatorname {adj} (\mathbf {A} )=\mathbf {C} ^{\mathsf {T}}={\begin{bmatrix}+{\begin{vmatrix}b_{2}&b_{3}\\c_{2}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{2}&a_{3}\\c_{2}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{2}&a_{3}\\b_{2}&b_{3}\end{vmatrix}}\\&&\\-{\begin{vmatrix}b_{1}&b_{3}\\c_{1}&c_{3}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{3}\\c_{1}&c_{3}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{3}\\b_{1}&b_{3}\end{vmatrix}}\\&&\\+{\begin{vmatrix}b_{1}&b_{2}\\c_{1}&c_{2}\end{vmatrix}}&-{\begin{vmatrix}a_{1}&a_{2}\\c_{1}&c_{2}\end{vmatrix}}&+{\begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}}\end{bmatrix}}.

3 × 3 numeric matrix[edit]

As a specific example, we have

\operatorname {adj} \!{\begin{bmatrix}-3&2&-5\\-1&0&-2\\3&-4&1\end{bmatrix}}={\begin{bmatrix}-8&18&-4\\-5&12&-1\\4&-6&2\end{bmatrix}}.

It is easy to check the adjugate is theinversetimes the determinant, $-6$ .

The $-1$ in the second row, third column of the adjugate was computed as follows. The (2,3) entry of the adjugate is the (3,2) cofactor ofA.This cofactor is computed using thesubmatrixobtained by deleting the third row and second column of the original matrixA,

{\begin{bmatrix}-3&-5\\-1&-2\end{bmatrix}}.

The (3,2) cofactor is a sign times the determinant of this submatrix:

(-1)^{3+2}\operatorname {det} \!{\begin{bmatrix}-3&-5\\-1&-2\end{bmatrix}}=-(-3\cdot -2--5\cdot -1)=-1,

and this is the (2,3) entry of the adjugate.

Properties[edit]

For any $n \times n$ matrix $A$ ,elementary computations show that adjugates have the following properties:

$\operatorname {adj} (\mathbf {I} )=\mathbf {I}$ ,where $\mathbf {I}$ is theidentity matrix.
$\operatorname {adj} (\mathbf {0} )=\mathbf {0}$ ,where $\mathbf {0}$ is thezero matrix,except that if $n=1$ then $\operatorname {adj} (\mathbf {0} )=\mathbf {I}$ .
$\operatorname {adj} (c\mathbf {A} )=c^{n-1}\operatorname {adj} (\mathbf {A} )$ for any scalar $c$ .
$\operatorname {adj} (\mathbf {A} ^{\mathsf {T}})=\operatorname {adj} (\mathbf {A} )^{\mathsf {T}}$ .
$\det(\operatorname {adj} (\mathbf {A} ))=(\det \mathbf {A} )^{n-1}$ .
IfAis invertible, then $\operatorname {adj} (\mathbf {A} )=(\det \mathbf {A} )\mathbf {A} ^{-1}$ $\operatorname {adj} (\mathbf {A} )=(\det \mathbf {A} )\mathbf {A} ^{-1}$ .It follows that:
- $adj(A)$ is invertible with inverse $(det A) -1 A$ .
- $adj(A -1) = adj(A) -1$ .
$adj(A)$ is entrywisepolynomialin $A$ .In particular, over therealor complex numbers, the adjugate is asmooth functionof the entries of $A$ .

Over the complex numbers,

$\operatorname {adj} ({\overline {\mathbf {A} }})={\overline {\operatorname {adj} (\mathbf {A} )}}$ ,where the bar denotescomplex conjugation.
$\operatorname {adj} (\mathbf {A} ^{*})=\operatorname {adj} (\mathbf {A} )^{*}$ ,where the asterisk denotesconjugate transpose.

Suppose that $B$ is another $n \times n$ matrix. Then

\operatorname {adj} (\mathbf {AB} )=\operatorname {adj} (\mathbf {B} )\operatorname {adj} (\mathbf {A} ).

This can beprovedin three ways. One way, valid for any commutative ring, is a direct computation using theCauchy–Binet formula.The second way, valid for the real or complex numbers, is to first observe that for invertible matrices $A$ and $B$ ,

\operatorname {adj} (\mathbf {B} )\operatorname {adj} (\mathbf {A} )=(\det \mathbf {B} )\mathbf {B} ^{-1}(\det \mathbf {A} )\mathbf {A} ^{-1}=(\det \mathbf {AB} )(\mathbf {AB} )^{-1}=\operatorname {adj} (\mathbf {AB} ).

Because every non-invertible matrix is the limit of invertible matrices,continuityof the adjugate then implies that the formula remains true when one of $A$ or $B$ is not invertible.

Acorollaryof the previous formula is that, for any non-negativeinteger $k$ ,

\operatorname {adj} (\mathbf {A} ^{k})=\operatorname {adj} (\mathbf {A} )^{k}.

If $A$ is invertible, then the above formula also holds for negative $k$ .

From the identity

(\mathbf {A} +\mathbf {B} )\operatorname {adj} (\mathbf {A} +\mathbf {B} )\mathbf {B} =\det(\mathbf {A} +\mathbf {B} )\mathbf {B} =\mathbf {B} \operatorname {adj} (\mathbf {A} +\mathbf {B} )(\mathbf {A} +\mathbf {B} ),

we deduce

\mathbf {A} \operatorname {adj} (\mathbf {A} +\mathbf {B} )\mathbf {B} =\mathbf {B} \operatorname {adj} (\mathbf {A} +\mathbf {B} )\mathbf {A}.

Suppose that $A$ commuteswith $B$ .Multiplying the identity $AB = BA$ on the left and right by $adj(A)$ proves that

\det(\mathbf {A} )\operatorname {adj} (\mathbf {A} )\mathbf {B} =\det(\mathbf {A} )\mathbf {B} \operatorname {adj} (\mathbf {A} ).

If $A$ is invertible, this implies that $adj(A)$ also commutes with $B$ .Over the real or complex numbers, continuity implies that $adj(A)$ commutes with $B$ even when $A$ is not invertible.

Finally, there is a more general proof than the second proof, which only requires that ann × nmatrix has entries over afieldwith at least 2n+ 1 elements (e.g. a 5 × 5 matrix over the integersmodulo11). $det(A + t I)$ is a polynomial intwithdegreeat mostn,so it has at mostnroots.Note that theij th entry of $adj((A + t I)(B))$ is a polynomial of at most ordern,and likewise for $adj(A + t I) adj(B)$ .These two polynomials at theij th entry agree on at leastn+ 1 points, as we have at leastn+ 1 elements of the field where $A + t I$ is invertible, and we have proven the identity for invertible matrices. Polynomials of degreenwhich agree onn+ 1 points must be identical (subtract them from each other and you haven+ 1 roots for a polynomial of degree at mostn– a contradiction unless their difference is identically zero). As the two polynomials are identical, they take the same value for every value oft.Thus, they take the same value whent= 0.

Using the above properties and other elementary computations, it is straightforward to show that if $A$ has one of the following properties, then $adj A$ does as well:

If $A$ isskew-symmetric,then $adj(A)$ is skew-symmetric for evennand symmetric for oddn.Similarly, if $A$ isskew-Hermitian,then $adj(A)$ is skew-Hermitian for evennand Hermitian for oddn.

If $A$ is invertible, then, as noted above, there is a formula for $adj(A)$ in terms of the determinant and inverse of $A$ .When $A$ is not invertible, the adjugate satisfies different but closely related formulas.

If $rk(A) \leq n - 2$ ,then $adj(A) = 0$ .
If $rk(A) = n - 1$ ,then $rk(adj(A)) = 1$ .(Some minor is non-zero, so $adj(A)$ is non-zero and hence hasrankat least one; the identity $adj(A) A = 0$ implies that thedimensionof thenullspaceof $adj(A)$ is at least $n - 1$ ,so its rank is at most one.) It follows that $adj(A) = α xy T$ ,where $α$ is a scalar and $x$ and $y$ are vectors such that $Ax = 0$ and $A T y = 0$ .

Column substitution and Cramer's rule[edit]

Partition $A$ intocolumn vectors:

\mathbf {A} ={\begin{bmatrix}\mathbf {a} _{1}&\cdots &\mathbf {a} _{n}\end{bmatrix}}.

Let $b$ be a column vector of size $n$ .Fix $1 \leq i \leq n$ and consider the matrix formed by replacing column $i$ of $A$ by $b$ :

(\mathbf {A} {\stackrel {i}{\leftarrow }}\mathbf {b} )\ {\stackrel {\text{def}}{=}}\ {\begin{bmatrix}\mathbf {a} _{1}&\cdots &\mathbf {a} _{i-1}&\mathbf {b} &\mathbf {a} _{i+1}&\cdots &\mathbf {a} _{n}\end{bmatrix}}.

Laplace expand the determinant of this matrix along column $i$ .The result is entry $i$ of the product $adj(A) b$ .Collecting these determinants for the different possible $i$ yields an equality of column vectors

\left(\det(\mathbf {A} {\stackrel {i}{\leftarrow }}\mathbf {b} )\right)_{i=1}^{n}=\operatorname {adj} (\mathbf {A} )\mathbf {b}.

This formula has the following concrete consequence. Consider thelinear system of equations

\mathbf {A} \mathbf {x} =\mathbf {b}.

Assume that $A$ isnon-singular.Multiplying this system on the left by $adj(A)$ and dividing by the determinant yields

\mathbf {x} ={\frac {\operatorname {adj} (\mathbf {A} )\mathbf {b} }{\det \mathbf {A} }}.

Applying the previous formula to this situation yieldsCramer's rule,

x_{i}={\frac {\det(\mathbf {A} {\stackrel {i}{\leftarrow }}\mathbf {b} )}{\det \mathbf {A} }},

where $x i$ is the $i$ th entry of $x$ .

Characteristic polynomial[edit]

Let thecharacteristic polynomialof $A$ be

p(s)=\det(s\mathbf {I} -\mathbf {A} )=\sum _{i=0}^{n}p_{i}s^{i}\in R[s].

The firstdivided differenceof $p$ is asymmetric polynomialof degree $n - 1$ ,

\Delta p(s,t)={\frac {p(s)-p(t)}{s-t}}=\sum _{0\leq j+k<n}p_{j+k+1}s^{j}t^{k}\in R[s,t].

Multiply $s I - A$ by its adjugate. Since $p (A) = 0$ by theCayley–Hamilton theorem,some elementary manipulations reveal

\operatorname {adj} (s\mathbf {I} -\mathbf {A} )=\Delta p(s\mathbf {I},\mathbf {A} ).

In particular, theresolventof $A$ is defined to be

R(z;\mathbf {A} )=(z\mathbf {I} -\mathbf {A} )^{-1},

and by the above formula, this is equal to

R(z;\mathbf {A} )={\frac {\Delta p(z\mathbf {I},\mathbf {A} )}{p(z)}}.

Jacobi's formula[edit]

The adjugate also appears inJacobi's formulafor thederivativeof the determinant. If $A (t)$ iscontinuously differentiable,then

{\frac {d(\det \mathbf {A} )}{dt}}(t)=\operatorname {tr} \left(\operatorname {adj} (\mathbf {A} (t))\mathbf {A} '(t)\right).

It follows that thetotal derivativeof the determinant is the transpose of the adjugate:

d(\det \mathbf {A} )_{\mathbf {A} _{0}}=\operatorname {adj} (\mathbf {A} _{0})^{\mathsf {T}}.

Cayley–Hamilton formula[edit]

Let $p A (t)$ be the characteristic polynomial of $A$ .TheCayley–Hamilton theoremstates that

p_{\mathbf {A} }(\mathbf {A} )=\mathbf {0}.

Separating the constant term and multiplying the equation by $adj(A)$ gives an expression for the adjugate that depends only on $A$ and the coefficients of $p A (t)$ .These coefficients can be explicitly represented in terms oftracesof powers of $A$ using complete exponentialBell polynomials.The resulting formula is

\operatorname {adj} (\mathbf {A} )=\sum _{s=0}^{n-1}\mathbf {A} ^{s}\sum _{k_{1},k_{2},\ldots,k_{n-1}}\prod _{\ell =1}^{n-1}{\frac {(-1)^{k_{\ell }+1}}{\ell ^{k_{\ell }}k_{\ell }!}}\operatorname {tr} (\mathbf {A} ^{\ell })^{k_{\ell }},

where $n$ is the dimension of $A$ ,and the sum is taken over $s$ and all sequences of $k l \geq 0$ satisfying the linearDiophantine equation

s+\sum _{\ell =1}^{n-1}\ell k_{\ell }=n-1.

For the 2 × 2 case, this gives

\operatorname {adj} (\mathbf {A} )=\mathbf {I} _{2}(\operatorname {tr} \mathbf {A} )-\mathbf {A}.

For the 3 × 3 case, this gives

\operatorname {adj} (\mathbf {A} )={\frac {1}{2}}\mathbf {I} _{3}\!\left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)-\mathbf {A} (\operatorname {tr} \mathbf {A} )+\mathbf {A} ^{2}.

For the 4 × 4 case, this gives

\operatorname {adj} (\mathbf {A} )={\frac {1}{6}}\mathbf {I} _{4}\!\left((\operatorname {tr} \mathbf {A} )^{3}-3\operatorname {tr} \mathbf {A} \operatorname {tr} \mathbf {A} ^{2}+2\operatorname {tr} \mathbf {A} ^{3}\right)-{\frac {1}{2}}\mathbf {A} \!\left((\operatorname {tr} \mathbf {A} )^{2}-\operatorname {tr} \mathbf {A} ^{2}\right)+\mathbf {A} ^{2}(\operatorname {tr} \mathbf {A} )-\mathbf {A} ^{3}.

The same formula follows directly from the terminating step of theFaddeev–LeVerrier algorithm,which efficiently determines thecharacteristic polynomialof $A$ .

In generally, adjugate matrix of arbitrary dimension N matrix can be computed by Einstein's convention.

(\operatorname {adj} (\mathbf {A} ))_{i_{N}}^{j_{N}}={\frac {1}{(N-1)!}}\epsilon _{i_{1}i_{2}\ldots i_{N}}\epsilon ^{j_{1}j_{2}\ldots j_{N}}A_{j_{1}}^{i_{1}}A_{j_{2}}^{i_{2}}\ldots A_{j_{N-1}}^{i_{N-1}}

Relation to exterior algebras[edit]

The adjugate can be viewed in abstract terms usingexterior algebras.Let $V$ be an $n$ -dimensionalvector space.Theexterior productdefines a bilinear pairing

V\times \wedge ^{n-1}V\to \wedge ^{n}V.

Abstractly, $\wedge ^{n}V$ isisomorphicto $R$ ,and under any such isomorphism the exterior product is aperfect pairing.Therefore, it yields an isomorphism

\phi \colon V\ {\xrightarrow {\cong }}\ \operatorname {Hom} (\wedge ^{n-1}V,\wedge ^{n}V).

Explicitly, this pairing sends $v \in V$ to $\phi _{\mathbf {v} }$ ,where

\phi _{\mathbf {v} }(\alpha )=\mathbf {v} \wedge \alpha.

Suppose that $T : V \to V$ is alinear transformation.Pullback by the $(n - 1)$ st exterior power of $T$ induces a morphism of $Hom$ spaces. Theadjugateof $T$ is the composite

V\ {\xrightarrow {\phi }}\ \operatorname {Hom} (\wedge ^{n-1}V,\wedge ^{n}V)\ {\xrightarrow {(\wedge ^{n-1}T)^{*}}}\ \operatorname {Hom} (\wedge ^{n-1}V,\wedge ^{n}V)\ {\xrightarrow {\phi ^{-1}}}\ V.

If $V = R n$ is endowed with itscanonical basis $e 1,\dots, e n$ ,and if the matrix of $T$ in thisbasisis $A$ ,then the adjugate of $T$ is the adjugate of $A$ .To see why, give $\wedge ^{n-1}\mathbf {R} ^{n}$ the basis

\{\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{k}\wedge \dots \wedge \mathbf {e} _{n}\}_{k=1}^{n}.

Fix a basis vector $e i$ of $R n$ .The image of $e i$ under $\phi$ is determined by where it sends basis vectors:

\phi _{\mathbf {e} _{i}}(\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{k}\wedge \dots \wedge \mathbf {e} _{n})={\begin{cases}(-1)^{i-1}\mathbf {e} _{1}\wedge \dots \wedge \mathbf {e} _{n},&{\text{if}}\ k=i,\\0&{\text{otherwise.}}\end{cases}}

On basis vectors, the $(n - 1)$ st exterior power of $T$ is

\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{j}\wedge \dots \wedge \mathbf {e} _{n}\mapsto \sum _{k=1}^{n}(\det A_{jk})\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{k}\wedge \dots \wedge \mathbf {e} _{n}.

Each of these terms maps to zero under $\phi _{\mathbf {e} _{i}}$ except the $k = i$ term. Therefore, the pullback of $\phi _{\mathbf {e} _{i}}$ is the linear transformation for which

\mathbf {e} _{1}\wedge \dots \wedge {\hat {\mathbf {e} }}_{j}\wedge \dots \wedge \mathbf {e} _{n}\mapsto (-1)^{i-1}(\det A_{ji})\mathbf {e} _{1}\wedge \dots \wedge \mathbf {e} _{n},

that is, it equals

\sum _{j=1}^{n}(-1)^{i+j}(\det A_{ji})\phi _{\mathbf {e} _{j}}.

Applying the inverse of $\phi$ shows that the adjugate of $T$ is the linear transformation for which

\mathbf {e} _{i}\mapsto \sum _{j=1}^{n}(-1)^{i+j}(\det A_{ji})\mathbf {e} _{j}.

Consequently, its matrix representation is the adjugate of $A$ .

If $V$ is endowed with aninner productand a volume form, then the map $φ$ can be decomposed further. In this case, $φ$ can be understood as the composite of theHodge star operatorand dualization. Specifically, if $ω$ is the volume form, then it, together with the inner product, determines an isomorphism

\omega ^{\vee }\colon \wedge ^{n}V\to \mathbf {R}.

This induces an isomorphism

\operatorname {Hom} (\wedge ^{n-1}\mathbf {R} ^{n},\wedge ^{n}\mathbf {R} ^{n})\cong \wedge ^{n-1}(\mathbf {R} ^{n})^{\vee }.

A vector $v$ in $R n$ corresponds to the linear functional

(\alpha \mapsto \omega ^{\vee }(\mathbf {v} \wedge \alpha ))\in \wedge ^{n-1}(\mathbf {R} ^{n})^{\vee }.

By the definition of the Hodge star operator, this linear functional is dual to $* v$ .That is, $ω \lor \circ φ$ equals $v \mapsto * v \lor$ .

Higher adjugates[edit]

Let $A$ be an $n \times n$ matrix, and fix $r \geq 0$ .The $r$ th higher adjugateof $A$ is an ${\textstyle {\binom {n}{r}}\!\times \!{\binom {n}{r}}}$ matrix, denoted $adj r A$ ,whose entries are indexed by size $r$ subsets $I$ and $J$ of ${1,..., m}$ ^{[citation needed]}.Let $I c$ and $J c$ denote thecomplementsof $I$ and $J$ ,respectively. Also let $\mathbf {A} _{I^{c},J^{c}}$ denote the submatrix of $A$ containing those rows and columns whose indices are in $I c$ and $J c$ ,respectively. Then the $(I, J)$ entry of $adj r A$ is

(-1)^{\sigma (I)+\sigma (J)}\det \mathbf {A} _{J^{c},I^{c}},

where $σ(I)$ and $σ(J)$ are the sum of the elements of $I$ and $J$ ,respectively.

Basic properties of higher adjugates include^{[citation needed]}:

$adj 0 (A) = det A$ .
$adj 1 (A) = adj A$ .
$adj n (A) = 1$ .
$adj r (BA) = adj r (A) adj r (B)$ .
$\operatorname {adj} _{r}(\mathbf {A} )C_{r}(\mathbf {A} )=C_{r}(\mathbf {A} )\operatorname {adj} _{r}(\mathbf {A} )=(\det \mathbf {A} )I_{\binom {n}{r}}$ ,where $C r (A)$ denotes the $r$ thcompound matrix.

Higher adjugates may be defined in abstract algebraic terms in a similar fashion to the usual adjugate, substituting $\wedge ^{r}V$ and $\wedge ^{n-r}V$ for $V$ and $\wedge ^{n-1}V$ ,respectively.

Iterated adjugates[edit]

Iterativelytaking the adjugate of an invertible matrixA $k$ times yields

\overbrace {\operatorname {adj} \dotsm \operatorname {adj} } ^{k}(\mathbf {A} )=\det(\mathbf {A} )^{\frac {(n-1)^{k}-(-1)^{k}}{n}}\mathbf {A} ^{(-1)^{k}},

\det(\overbrace {\operatorname {adj} \dotsm \operatorname {adj} } ^{k}(\mathbf {A} ))=\det(\mathbf {A} )^{(n-1)^{k}}.

For example,

\operatorname {adj} (\operatorname {adj} (\mathbf {A} ))=\det(\mathbf {A} )^{n-2}\mathbf {A}.

\det(\operatorname {adj} (\operatorname {adj} (\mathbf {A} )))=\det(\mathbf {A} )^{(n-1)^{2}}.

References[edit]

^Gantmacher, F. R.(1960).The Theory of Matrices.Vol. 1. New York: Chelsea. pp. 76–89.ISBN 0-8218-1376-5.
^Strang, Gilbert(1988)."Section 4.4: Applications of determinants".Linear Algebra and its Applications(3rd ed.). Harcourt Brace Jovanovich. pp.231–232.ISBN 0-15-551005-3.
^Claeyssen, J.C.R. (1990). "On predicting the response of non-conservative linear vibrating systems by using dynamical matrix solutions".Journal of Sound and Vibration.140(1): 73–84.Bibcode:1990JSV...140...73C.doi:10.1016/0022-460X(90)90907-H.
^Chen, W.; Chen, W.; Chen, Y.J. (2004). "A characteristic matrix approach for analyzing resonant ring lattice devices".IEEE Photonics Technology Letters.16(2): 458–460.Bibcode:2004IPTL...16..458C.doi:10.1109/LPT.2003.823104.
^Householder, Alston S.(2006).The Theory of Matrices in Numerical Analysis.Dover Books on Mathematics. pp. 166–168.ISBN 0-486-44972-6.

Bibliography[edit]

Roger A. Horn and Charles R. Johnson (2013),Matrix Analysis,Second Edition. Cambridge University Press,ISBN 978-0-521-54823-6
Roger A. Horn and Charles R. Johnson (1991),Topics in Matrix Analysis.Cambridge University Press,ISBN 978-0-521-46713-1

External links[edit]

Matrix Reference Manual
Online matrix calculator (determinant, track, inverse, adjoint, transpose)Compute Adjugate matrix up to order 8
"Adjugate of { { a, b, c }, { d, e, f }, { g, h, i } }".Wolfram Alpha.

[1] Gantmacher, F. R.(1960).The Theory of Matrices.Vol. 1. New York: Chelsea. pp. 76–89.ISBN 0-8218-1376-5.

[2] Strang, Gilbert(1988)."Section 4.4: Applications of determinants".Linear Algebra and its Applications(3rd ed.). Harcourt Brace Jovanovich. pp.231–232.ISBN 0-15-551005-3.

[3] Claeyssen, J.C.R. (1990). "On predicting the response of non-conservative linear vibrating systems by using dynamical matrix solutions".Journal of Sound and Vibration.140(1): 73–84.Bibcode:1990JSV...140...73C.doi:10.1016/0022-460X(90)90907-H.

[4] Chen, W.; Chen, W.; Chen, Y.J. (2004). "A characteristic matrix approach for analyzing resonant ring lattice devices".IEEE Photonics Technology Letters.16(2): 458–460.Bibcode:2004IPTL...16..458C.doi:10.1109/LPT.2003.823104.

[5] Householder, Alston S.(2006).The Theory of Matrices in Numerical Analysis.Dover Books on Mathematics. pp. 166–168.ISBN 0-486-44972-6.

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Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions oneigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz Positive-definite Stieltjes
Satisfying conditions onproductsorinverses	Congruent IdempotentorProjection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation
Used instatistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used ingraph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
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