Download ON THE CHARACTERISTIC POLYNOMIAL OF THE PRODUCT OF

ON THE CHARACTERISTIC POLYNOMIAL OF THE
PRODUCT OF SEVERAL MATRICES
WILLIAM E. ROTH
We shall prove two theorems.
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 poly-
nomial of Ai = A-\-Diis
\xl
— Ai\
= mi0 + 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)\,
'»»lo,
(1)
A(x') =
where
Wi,r_iXr-1,
OTi,r_2Xr~2,
• • • , »»llX
»»2lX,
»»20,
»»2,r-lXr~1,
• • • , »»22X2
»»32X2,
»»3lX,
»»30,
• • • , 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¡ = 0,
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, 0, • • • , 0)r; as a result
QDiQ' = (1, 0, • • • , 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
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579
PRODUCT OF SEVERAL MATRICES
(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
Nk(x)
[xi - Mk][xl - M]A,
mk(x)
*
*
■■
0
m(x)
0
•■
0
0
0
0
(3)
k = \, 2, • • • , r, where asterisks
m(x) ■ ■
0
• m(x)j
indicate
nonzero elements
in F(x)
and \xl —M\ =m(x).
Let
W = («,,) - («<«-»<'-»),
i,j=
1,2, •
r;
then
W(h) | = | W |
(4)
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
(«r+1,1 = 1)
0
•
- M2 ■ ■ ■
0
0
xi
0
0
0
Mr-l
Mr
0
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
0, /, M2x~l, • • • , MiMz ■ ■ ■ Mr-iX~r+2, and whose last row is
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580
W. E. ROTH
[August
0, 0, 0, ■ • • , 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)(-«-|
We now proceed to establish
A(0| .
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
the matric equation
(6)
Tne
determinantal
equation
obtained
from
above is as a result:
| 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«)),
0,
l
o,
*
(uiMa'^x)),
o
, •••,
*
■ ■ ■,
0
, • • •, (¡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
| uijiu'^xl
- Mi] | [ n»»(«f'"1ï)]""1
= i w i«-1[ n«(«Hï)
]n_i i (coijWiO''-1*))i,
or
(9)
| œalco'-'xl- Mi] | = \W I"-11(ttyWifw*-1*))
|,
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581
PRODUCT OF SEVERAL MATRICES
1956]
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. Here-
with equation
(5) is established
and the proof of Theorem
I is com-
pleted.
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 +
then the characteristic
polynomial
( —l)(r-»»|A'(a;)|
where
• BT is
of BiB2
Mr-1,1*,
Mr_i,o,
[Ml,,-!*'"
Ml,r-2*r'
A'(*0 =
• ■ • + «¿,r-i£r_1;
given
by
• ■ • , Mr-1,2*2
, Mi,0
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 fol-
lows that in A(xr) of (1) we must replace the elements íM¿,y_iXí_1by
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
(10)
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) =
mr-ixr'
Mr-iX"-2,
■, mix
mix,
Wo,
mT-iXr~x,
■ , m2x2
m2x2,
m,\x,
mo,
• , m3x3
mr-2xr
2, mr-3xr~i,
\mr^iXT l,
According to a theorem by Frobenius
• • • ,mo
J
[4], the determinant
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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 (10) 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
| xl — PiA'-'l = | xl - Ar\ ;
then
| 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
| xl - Pi+iA'-i-11 = | xl - PiAr-i | = | xl - A' | .
Consequently
\xl - P\ = | xl - A' | = (-l)c-D»
and the theorem
| 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. 299-300.
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. 601-614.
5. Schneider,
A pair of matrices having the property P, Amer. Math.
vol. 62 (1955) pp. 247-249.
Mississippi City, Miss.
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