On the adjugate of a matrix

2 Pages·0.064 MB·English

by  António Guedes de Oliveira

On the adjugate of a matrix
Anto(cid:19)nio Guedes de Oliveira(cid:3)
Universidade do Porto
Let j(cid:21)I (cid:0)Aj = (cid:21)n +cn(cid:0)1(cid:21)n(cid:0)1 +(cid:1)(cid:1)(cid:1)+c1(cid:21)+c0 be the characteristic polynomial of an
n-by-n matrix A over a given (cid:12)eld K. The elegant proof of the Cayley-Hamilton theorem of
[1, p. 50] can be easily modi(cid:12)ed to prove that (see [2, p. 40]):
((cid:0)1)n(cid:0)1Aadj = An(cid:0)1 +cn(cid:0)1An(cid:0)2 +(cid:1)(cid:1)(cid:1)+c1I (1)
where Aadj stands for the adjugate of A (or classical adjoint | the transpose of the cofactor
matrixofA)andI fortheidentitymatrixofordern. Moregenerally, itcanbeeasilymodi(cid:12)ed
to prove that (see [3, p. 38]):
((cid:21)I(cid:0)A)adj = An(cid:0)1+((cid:21)+cn(cid:0)1)An(cid:0)2+(cid:1)(cid:1)(cid:1)+((cid:21)n(cid:0)1+cn(cid:0)1(cid:21)n(cid:0)2+cn(cid:0)2(cid:21)n(cid:0)3+(cid:1)(cid:1)(cid:1)+c1)I (2)
Proof. We start from the basic fact that:
((cid:21)I (cid:0)A)(cid:1)((cid:21)I (cid:0)A) adj = ((cid:21)n+cn(cid:0)1(cid:21)n(cid:0)1+(cid:1)(cid:1)(cid:1)+c1(cid:21)+c0)I (3)
and by noting that, by de(cid:12)nition of adjugate, ((cid:21)I (cid:0) A) adj is a polynomial in (cid:21) of degree
n(cid:0)1 with coe(cid:14)cients in the space of the n-by-n matrices over K, say,
((cid:21)I (cid:0)A)adj = Dn(cid:0)1(cid:21)n(cid:0)1+Dn(cid:0)2(cid:21)n(cid:0)2+(cid:1)(cid:1)(cid:1)+D1(cid:21)+D0
By equating the terms in (cid:21) of the same order in both sides of equation (3), we obtain:
Dn(cid:0)1 = I
(cid:0)A(cid:1)Dn(cid:0)1 +Dn(cid:0)2 = cn(cid:0)1I
(cid:0)A(cid:1)Dn(cid:0)2 +Dn(cid:0)3 = cn(cid:0)2I
. . .
. . .
. . .
(cid:0)A(cid:1)D +D = c I
1 0 1
(cid:0)A(cid:1)D = c I
0 0
Finally, by multiplying the (cid:12)rst equation by An, the second by An(cid:0)1, and so on, up to the
last equation, and by adding the new equations up, we obtain the Cayley-Hamilton theorem.
This is the proof in [1, p. 50]. If, instead, we multiply the (cid:12)rst equation by An(cid:0)1, the second
by An(cid:0)2, etc., and stop precisely before the last equation, by summing up we obtain at the
left-hand side D , which is ((cid:0)1)n(cid:0)1Aadj, say, by putting (cid:21)= 0. Hence, we get (1). (2) can be
0
proven by obtaining the coe(cid:14)cients of ((cid:21)I(cid:0)A) adj step by step through the same procedure.
(cid:3)
Work partially supported by the Centro de Matematica da Universidade do Porto (CMUP), (cid:12)nanced
by FCT (Portugal) through the programmes POCTI and POSI, with national and European Community
structural funds.
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References
[1] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Dover
Publications, Inc, New York (1992).
[2] O. Taussky, \The factorization of the adjugate of a (cid:12)nite matrix", Linear Algebra Appl.,
1 (1968), pp. 39{41.
[3] R.A. Frazer, W.J. Duncan, and A.R. Collar, Elementary Matrices and Some Applica-
tions to Dynamics and Di(cid:11)erential Equations, Cambridge University Press, Cambridge,
England, (1955).
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