ON THE CHARACTERISTIC POLYNOMIAL OF THE PRODUCT OF SEVERAL MATRICES We shall prove two theorems. WILLIAM E. ROTH Theorem I. If A is an nxn matrix with elements in the field F, if R and 5<, * = 1, 2,, r, are IX«matrices with elements in F, and Di==RTSi, where RT is the transpose of R, and if the characteristic polynomial of Ai = A-\-Diis \xl Ai\ = mi + max + mi2x2 + + mi,r-ixr~1, where w.-,y_i, i, j=\, 2,, r, are polynomials in xr with coefficients in F, then the characteristic polynomial of the product P=AiA2 AT is given by { l){r~1)n\a{x)\, where '»»lo, Wi,r_iXr-1, OTi,r_2Xr~2,,»»llX (1) A(x') =»»2lX,»»2,»»2,r-lXr~1,,»»22X2»»32X2,»»3lX,»»3,, W33X3 (»»r,r_lxr-1,»»rir_2xr 2,»»r,r_3Xr 3,,»»r,o j This proposition has been proved by the writer [l] for the case r = 2. Recently Parker [2] generalized that result and Goddard [3] gave an alternate proof of it and extended his method to the product of three matrices. This latter result does not come under the theorem above. Schneider [5] proved the theorem for the case AiA =, i<j, i,j=l, 2,, r. Capital letters and expressions in bold faced parentheses will indicate matrices with elements in the field F or in F(w), the extension of F by the adjunction of «a primitive rth root of unity to it, and in F{x) the polynomial domain of F{u). The direct product of B and C is {bijc)=b(c). The product indicated by LI will run from 1 to r. If R is not zero a nonsingular matrix Q with elements in F exists such that QRT = {1,,, )r; as a result QDiQ' = (1,,, OySiQ1 = Ei, where Q1 is the inverse of Q and, has nonzero elements in only the first row. Now let Presented to the Society, April 23, 1955; received by the editors April 4, 1955 and, in revised form, August 26, 1955. 578
PRODUCT OF SEVERAL MATRICES 579 (2) QAkQ> = Mk = Q(A + Dk)Q' = M + Ek, where Q is the matrix defined above and QAQ1 = M=(mn). Consequently Mk = (m\f), where mf =mij-\-ef] and m\f =ma for *>1. That is, the matrices Mk differ only in the elements of their first rows. As a result the elements of the first columns of the adjoints [xi Mk]A and [xi M]A of xi Mk and xi M respectively are identical for k = 1, 2,, r, since all these matrices agree in the elements of their last n 1 rows and for the same reason (3) Nk(x) [xi - Mk][xl - M]A, mk(x) * * m(x) m(x) m(x)j k = \, 2,, r, where asterisks indicate nonzero elements in F(x) and \xl M\ =m(x). Let W = («,,) - («<«-»<'-»), i,j= 1,2, r; then (4) W(h) = W where Ik is the identity matrix of order k. The determinantal equation holds because W(Ik) can be transformed by the interchange of rows and corresponding columns to the direct sum W-j-W-j- -\-W of k summands. We shall operate in F(x) on the matrix M(x) = (dijxl 5i+i,jMi) xi - Mi xi - M2 («r+1,1 = 1) Mr Mr-l xi If we multiply this matrix on the right by one whose first row is /, Mix~l, MiM2x~2,, MiM2 MT-ix~r+1, whose second row is, /, M2x~l,, MiMz Mr-iX~r+2, and whose last row is
58 W. E. ROTH [August,,,, i", we find that the determinant of the product is \xri MTMiM2 MT-i\ and is therefore equal to \xr P\. The proof of Theorem I will consist in showing that (5) M{x)\ = (-i)(-«- A(. We now proceed to establish this equation. M{x)W{I) = {taxi - ôi+i,kmi){^-^-'h), = («Cí-»a-%j _ uhi-fími), = (^-'{wiiiw'-ixl - Mi]}). The number co1-' is a common multiplier of the nxn matrices in the 7th column of the nr Xnr matrix in right member above. Consequently the determinant of this matrix has the factor 7rtu(1-*)n=ûr"r(r~1)"'î =wr(r-i)n/2 = (_i)(r-i)n Tne determinantal equation obtained from the matric equation above is as a result: (6) M{x) W " = (-l)c^y» Uijiu'-ixl - Mi]. According to (3) the product (7) {wiklco^xl - Mi\){hjW-lxI - M]A) ={uijni{ui-1x)). The nrxnr matrix of the right member of this equation can be transformed by the interchange of corresponding rows and columns to a similar one having the form Xuijmiia*-1«)), *,, *, (uima'^x)),, l o, o,, ( tumft~1*)) J where asterisks represent rxr matrices with elements in F{x) and the zeros are rxr zero matrices. The determinant of this matrix is (8) \W I""1 [Hm^'-'x)Y'11 (wyiäicy-1*)) I for each of the matrices (wij»w(coi_1x)) has»w(a>i-1x) as a divisor of all elements in the jth column. The determinant of the direct sum {8ij[w'~1xl M]A) in equation (7) is [ [Jw(co/_1x)]n_1; consequently the determinantal equation which follows from (7) and (8) is or uijiu'^xl - Mi] [ n»»(«f'"1ï)]""1 = i w i«-1 [ n«(«hï) ]n_i i (coijwio''-1*)) i, (9) œalco'-'xl - Mi] = \W I"-11 (ttywifw*-1*)),
1956] PRODUCT OF SEVERAL MATRICES 581 where the determinant of the left member is that of an nrxnr matrix and those in the right members are of order r. From (6) and (9) we have \M(x)\=(-l)^-l)n\(uijMi(u'-1x))\/\W\. It remains to be shown that the right member here is ( l)(r-1)" A(xr) ; this is easily accomplished by multiplying A(xr) in (1) on the right by W. Herewith equation (5) is established and the proof of Theorem I is completed. Corollary. Under the hypotheses of Theorem I and if B is an nxn matrix with elements in F and if Bi = B-\-SjR and \ xi Bi\ = «o + nnx + ni2x2 + + «,r-i r_1; then the characteristic polynomial of BiB2 ( l)(r-»» A'(a;) where BT is given by A'(* = Mr-1,1*, Mr_i,o,, Mr-1,2*2 [Ml,,-!*'" Ml,r-2*r', Mi, This case can be made to come under Theorem I for Bf = BT+RTSi, where Bf now satisfies the conditions imposed upon A{. Moreover \xi-bi\ =\xi-bf\. Since (B,B, Br)T = BfB?_1 Bf, it follows that in A(xr) of (1) we must replace the elements ím,y_ixí_1 by Mr-i+i.y-ix'-1 in forming the matrix A'(*r) above. This proves the corollary. Theorem II. If Dit i = i, 2,, r, arenxn matrices with elements in F, each of which is nilpotent and commutative with the others and with A, which also has elements in F, then the characteristic polynomials of Ai = A-\-Di, i = l, 2,, r, are given by (1) xi A\ = trio + mix + m2x2 + + ntr-ix*-1, where wt-_i, * = 1, 2,, r, are polynomials in xr with coefficients in F, and the characteristic polynomial of the product P=AiA2 Ar is (_l)(r-i)n Â"(x), where A(xr) = mix, m2x2, mr-ixr' Wo, m,\x, Mr-iX"-2, mt-ixr~x, mo,, mix, m2x2, m3x3 \mr^ixt l, mr-2xr 2, mr-3xr~i,,mo J According to a theorem by Frobenius [4], the determinant of the
582 W. E. ROTH matrix B-\-C is equal to that of B if B and C are commutative matrices and C is nilpotent. This establishes equation (1) as giving the characteristic polynomial of Ai, * = 1, 2,, r. By Theorem I the determinant xl - A*\ = (-l)fr-»» ÄO). We shall proceed by induction. Let Pi=AiA2 -Ai, then xl - PiA'-11 = xl - {A + Di)Ar-11 = xl - Ar - DiA*-1 \. Now the matrix D Ar~l is nilpotent and commutative with xi AT consequently the determinants above are equal to \xl AT\. We assume that then xl PiA'-'l = xl - Ar\ ; xl - Pi+iA'-i-11 = xl - PiA^-i - PiDi+iAr-i-11. Here PiZ>i+i^4r_i_1 is commutative with xi PiA'^ and is nilpotent because Di+i is nilpotent and commutative with both P, and A ; hence by Frobenius' theorem Consequently and the theorem xl - Pi+iA'-i-11 = xl - PiAr-i = xl - A'. \xl - P\ = xl - A' = (-l)c-d» A{x), is proved. References 1. W. E. Roth, On the characteristic polynomial of the product of two matrices, Proc. Amer. Math. Soc. vol. 5 (1954) pp. 1-3. 2. W. V. Parker, A note on a theorem of Roth, Proc. Amer. Math. Soc. vol. 6 (1955) pp. 299-3. 3. L. S. Goddard, On the characteristic function of a matrix product, Proc. Amer. Math. Soc. vol. 6 (1955) pp. 296-298. 4. Frobenius, Preuss. Akad. Wiss. Sitzungsber. (1896) I pp. 61-614. 5. Schneider, A pair of matrices having the property P, Amer. Math. Monthly vol. 62 (1955) pp. 247-249. Mississippi City, Miss.