Page 107 - Textos de Matemática Vol. 44

SPECTRUM OF POLYNOMIALS OF TWO MATRICES 97
In these notations polynomials F(A,B) and F(B,A) can be written in the “vector form” :
(3.3) F(A,B)= a C , F(B,A)= a C = a C . pppppp
Let Sm denote the set of monomials Cp = Cp1 Cp2 ...Cpm , and N the set of natural numbers. For given m ∈ N and l (0 ≤ l ≤ m) we denote by Sm(l) the set of monomials Cp ∈ Sm which contain l factors A and r = m − l factors B. The next proposition gives a useful necessary condition.
Proposition 3.2. Let F(A,B) ∈ Pm and let n ∈ N. If trF(A,B) = trF(B,A) for all A,B ∈ Mn, then
(3.4) ap= ap Cp ∈Sk (l) Cp ∈Sk (l)
for each k (1 ≤ k ≤ m) and each l (0 ≤ l ≤ k).
Proof. Let A,B be the following two scalar matrices: A = xE, B = yE.
Then (3.5)
where (3.6)
mm trF(A,B) = blxlyr, trF(B,A) = dlxlyr,
l=0 l=0
bl= ap, dl= ap. Cp ∈Sm (l) Cp ∈Sm (l)
Since tr F (A, B) = tr F (B, A), the equality (3.4) follows from (3.5) and (3.6).
Example 3.3. Let F ∈ P4 and n ∈ N. Equality (3.4) holds for any two matrices A, B ∈ Mn if and only if
a0 = a1; a00 = a11; a000 = a111; a0000 = a1111; (3.7) a001 + a010 + a100 = a110 + a101 + a011;
a0001 + a0010 + a0100 + a1000 = a1110 + a1101 + a1011 + a0111. Example 3.4. Let F(A,B) ∈ P2. and trF(A,B) = trF(B,A). It follows from
(3.7) that a00 = a11 and a0 = a1, i.e.,
(3.8) F(A,B)=a(A2 +B2)+bAB+cBA+d(A+B)+qE.
Note that in this case tr F (A, B) − tr(B, A) = (b − c) tr(AB − BA) = 0, i.e. the condition (3. 4) turns out to be sufficient, too.