Adjugate matrix

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In linear algebra, the adjugate, classical adjoint, or adjunct of a square matrix is the transpose of its cofactor matrix.[1]

The adjugate[2] has sometimes been called the "adjoint",[3] but today the "adjoint" of a matrix normally refers to its corresponding adjoint operator, which is its conjugate transpose.

Definition[edit]

The adjugate of A is the transpose of the cofactor matrix C of A,

In more detail, suppose R is a commutative ring and A is an n × n matrix with entries from R. The (i,j)-minor of A, denoted Mij, is the determinant of the (n − 1) × (n − 1) matrix that results from deleting row i and column j of A. The cofactor matrix of A is the n × n matrix C whose (i, j) entry is the (i, j) cofactor of A, which is the (i, j)-minor times a sign factor:

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

The adjugate is defined as it is so that the product of A with its adjugate yields a diagonal matrix whose diagonal entries are the determinant det(A). That is,

where I is the n×n identity matrix. This is a consequence of the Laplace expansion of the determinant.

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

Examples[edit]

1 × 1 generic matrix[edit]

The adjugate of any 1×1 matrix is .

2 × 2 generic matrix[edit]

The adjugate of the 2×2 matrix

is

By direct computation,

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

Its cofactor matrix is

where

.

Its adjugate is the transpose of its cofactor matrix,

.

3 × 3 numeric matrix[edit]

As a specific example, we have

It is easy to check the adjugate is the inverse times 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 of A. This cofactor is computed using the submatrix obtained by deleting the third row and second column of the original matrix A,

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

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

Properties[edit]

For any n × n matrix A, elementary computations show that adjugates enjoy the following properties.

  • and , where and are the zero and identity matrices, respectively.
  • for any scalar c.
  • .
  • .
  • If A is invertible, then . It follows that:
    • adj(A) is invertible with inverse (det A)−1 A.
    • adj(A−1) = adj(A)−1.
  • adj(A) is entrywise polynomial in A. In particular, over the real or complex numbers, the adjugate is a smooth function of the entries of A.

Over the complex numbers,

  • , where the bar denotes complex conjugation.
  • , where the asterisk denotes conjugate transpose.

Suppose that B is another n × n matrix. Then

This can be proved in two ways. One way, valid for any commutative ring, is a direct computation using the Cauchy–Binet formula. The other way, valid for the real or complex numbers, is to first observe that for invertible matrices A and B,

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

A corollary of the previous formula is that, for any non-negative integer k,

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

From the identity

we deduce

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

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.

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:

  • Upper triangular,
  • Lower triangular,
  • Diagonal,
  • Orthogonal,
  • Unitary,
  • Symmetric,
  • Hermitian,
  • Skew-symmetric,
  • Skew-hermitian,
  • Normal.

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) ≤ 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 has rank at least one; the identity adj(A) A = 0 implies that the dimension of the nullspace of adj(A) is at least n − 1, so its rank is at most one.) It follows that adj(A) = αxyT, where α is a scalar and x and y are vectors such that Ax = 0 and ATy = 0.

Column substitution and Cramer's rule[edit]

Partition A into column vectors:

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

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

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

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

Applying the previous formula to this situation yields Cramer's rule,

where xi is the ith entry of x.

Characteristic polynomial[edit]

Let the characteristic polynomial of A be

The first divided difference of p is a symmetric polynomial of degree n − 1,

Multiply sIA by its adjugate. Since p(A) = 0 by the Cayley–Hamilton theorem, some elementary manipulations reveal

In particular, the resolvent of A is defined to be

and by the above formula, this is equal to

Jacobi's formula[edit]

The adjugate also appears in Jacobi's formula for the derivative of the determinant. If A(t) is continuously differentiable, then

It follows that the total derivative of the determinant is the transpose of the adjugate:

Cayley–Hamilton formula[edit]

Let pA(t) be the characteristic polynomial of A. The Cayley–Hamilton theorem states that

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 pA(t). These coefficients can be explicitly represented in terms of traces of powers of A using complete exponential Bell polynomials. The resulting formula is

where n is the dimension of A, and the sum is taken over s and all sequences of kl ≥ 0 satisfying the linear Diophantine equation

For the 2×2 case, this gives

For the 3×3 case, this gives

For the 4×4 case, this gives

The same formula follows directly from the terminating step of the Faddeev–LeVerrier algorithm, which efficiently determines the characteristic polynomial of A.

Relation to exterior algebras[edit]

The adjugate can be viewed in abstract terms using exterior algebras. Let V be an n-dimensional vector space. The exterior product defines a bilinear pairing

Abstractly, is isomorphic to R, and under any such isomorphism the exterior product is a perfect pairing. Therefore, it yields an isomorphism

Explicitly, this pairing sends vV to φv, where

Suppose that T : VV is a linear transformation. Pullback by the (n − 1)st exterior power of T induces a morphism of Hom spaces. The adjugate of T is the composite

If V = Rn is endowed with its coordinate basis e1, ..., en, and if the matrix of T in this basis is A, then the adjugate of T is the adjugate of A. To see why, give the basis

Fix a basis vector ei of Rn. The image of ei under φ is determined by where it sends basis vectors:

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

Each of these terms maps to zero under except the k = i term. Therefore, the pullback of is the linear transformation for which

that is, it equals

Applying the inverse of φ shows that the adjugate of T is the linear transformation for which

Consequently, its matrix representation is the adjugate of A.

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

This induces an isomorphism

A vector v in Rn corresponds to the linear functional

By the definition of the Hodge star operator, this linear functional is dual to *v. That is, ω ∘ φ equals v ↦ *v.

Higher adjugates[edit]

Let A be an n × n matrix, and fix r ≥ 0. The rth higher adjugate of A is an matrix, denoted adjr A, whose entries are indexed by size r subsets I and J of {1, ..., m}. Let Ic and Jc denote the complements of I and J, respectively. Also let denote the submatrix of A containing those rows and columns whose indices are in Ic and Jc, respectively. Then the (I, J) entry of adjr A is

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

Basic properties of higher adjugates include:

  • adj0(A) = det A.
  • adj1(A) = adj A.
  • adjn(A) = 1.
  • adjr(BA) = adjr(A) adjr(B).
  • , where Cr(A) denotes the rth compound matrix.

Higher adjugates may be defined in abstract algebraic terms in a similar fashion to the usual adjugate, substituting and for and , respectively.

Iterated adjugates[edit]

Iteratively taking the adjugate of an invertible matrix A k times yields

For example,

See also[edit]

References[edit]

  1. ^ Gantmacher, F. R. (1960). The Theory of Matrices. 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. ^ Householder, Alston S. (2006). The Theory of Matrices in Numerical Analysis. Dover Books on Mathematics. pp. 166–168. ISBN 0-486-44972-6.
  • 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]