adjugate
The adjugate, , of an matrix , is the matrix
(1) |
where is the indicated minor of (the determinant obtained by deleting row and column from ). The adjugate is also known as the classical adjoint, to distinguish it from the usual usage of “adjoint” (http://planetmath.org/AdjointEndomorphism) which denotes the conjugate transpose operation.
An equivalent characterization of the adjugate is the following:
(2) |
The equivalence of (1) and (2) follows easily from the multi-linearity properties (http://planetmath.org/DeterminantAsAMultilinearMapping) of the determinant. Thus, the adjugate operation is closely related to the matrix inverse. Indeed, if is invertible, the adjugate can be defined as
Yet another definition of the adjugate is the following:
(3) | ||||
where are the elementary invariant polynomials of . The latter arise as coefficients in the characteristic polynomial of , namely
The equivalence of (2) and (3) follows from the Cayley-Hamilton theorem. The latter states that , which in turn implies that
The adjugate operation enjoys a number of notable properties:
(4) | |||
(5) | |||
(6) |
Title | adjugate |
---|---|
Canonical name | Adjugate |
Date of creation | 2013-03-22 13:09:42 |
Last modified on | 2013-03-22 13:09:42 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 17 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 15A09 |
Synonym | classical adjoint |