Adjugate matrix
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
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,
- 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 det(A),
A is invertible if and only if det(A) is an invertible element of R, and in that case the equation above yields
Examples
1 × 1 generic matrix
The adjugate of any 1×1 matrix is .
2 × 2 generic matrix
The adjugate of the 2×2 matrix
is . It is seen that det(adj(A)) = det(A) and hence that adj(adj(A)) = A.
3 × 3 generic matrix
Consider a 3×3 matrix
is
where
- .
Its adjugate is the transpose of its cofactor matrix:
3 × 3 numeric matrix
As a specific example, we have
- .
The −6 in the third row, second column of the adjugate was computed as follows:
Again, the (3,2) entry of the adjugate is the (2,3) cofactor of A. Thus, the submatrix
was obtained by deleting the second row and third column of the original matrix A.
It is easy to check the adjugate is the inverse times the determinant, −6.
Properties
The adjugate has the properties
for n×n matrices A and B. The second line follows from equations adj(B)adj(A) = det(B)B−1 det(A)A−1 = det(AB)(AB)−1.
Substituting in the second line B = Am − 1 and performing the recursion, one finds, for all integer m,
The adjugate preserves transposition,
Furthermore,
- If A is a n×n matrix with n ≥ 2, then and
- If A is an invertible n×n matrix, then
so that, if n = 2 and A is invertible, then det(adj(A)) = det(A) and adj(adj(A)) = A.
Taking the adjugate of an invertible matrix A k times yields
Inverses
In consequence of Laplace's formula for the determinant of an n×n matrix A,
where is the n×n identity matrix. Indeed, the (i,i) entry of the product A adj(A) is the scalar product of row i of A with row i of the cofactor matrix C, which is simply the Laplace formula for det(A) expanded by row i.
Moreover, for i ≠ j the (i,j) entry of the product is the scalar product of row i of A with row j of C, which is the Laplace formula for the determinant of a matrix whose i and j rows are equal, and therefore vanishes.
From this formula follows one of the central results in matrix algebra: A matrix A over a commutative ring R is invertible if and only if det(A) is invertible in R.
For, if A is an invertible matrix, then
and equation (*) above implies
Similarly, the resolvent of A is
where p(t) is the characteristic polynomial of A.
Characteristic polynomial
If
is the characteristic polynomial of the matrix n-by-n matrix with coefficients in the ring R, then
where
is the first divided difference of p, a symmetric polynomial of degree n−1.
Jacobi's formula
The adjugate also appears in Jacobi's formula for the derivative of the determinant,
Cayley–Hamilton formula
The Cayley–Hamilton theorem allows the adjugate of A to be represented in terms of traces and powers of A:
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 results follow directly from the terminating step of the fast Faddeev–LeVerrier algorithm.
See also
References
- ↑ Gantmacher, F. R. (1960). The Theory of Matrices. 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.
- ↑ 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 (1991), Topics in Matrix Analysis. Cambridge University Press, ISBN 978-0-521-46713-1
External links
- 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.