Page 111 - Textos de Matemática Vol. 44

SPECTRUM OF POLYNOMIALS OF TWO MATRICES 101
the polynomial F1. The system of equations {ck = 0, k = 0, 1, 2, 3} gives the solution
b+c (b−c)2
(5.3) a=− 2 , d=− 16r , k=−r, m=−r.
If r ̸= 0, then m ̸= 0. We can set m = 1. Then, taken into account (5.3) and equality n = r, proved just above, we obtain that the polynomial (5.1) coincides, up to multiplication by the constant m, with the polynomial (1.4).
If r = 0, then, taking into account that n = r and making similar manipula- tions with the functions f1(x) and g1(x) as it was done above, we obtain that the admissible non-trivial polynomial F1 satisfies the following conditions.
(5.4) m=0, a=0, k̸=0 d= bc. Onemayassumethatk=1. 2k
Now we replace the matrix A with the matrix A0 = 2E11 + E21 − E23 + E33 and keep same matrix B. We obtain that 0 = tr F1(A0, B)3 − tr F1(B, A0)3 = 9(b − c) ̸= 0. Thus r = 0 gives nothing additional to the case r ̸= 0.
Next we have to consider the polynomial F2. First we see that 0 = f2(−1) = 4(r − n)(2k − n − r). If n = r, then the polynomials F1 and F2 coincide, and we do not need to consider it again. It remains that 2k = n + r and we will take this into the consideration. Now it is convenient to consider two 4 × 4 matrices A := diag(1, 1, x, x2) and B := E12 + E23 + E34 + E41. For these matrices g2(x) = h2(x) 4k=0 dkxk. From the system of equations { dk = 0, k = 0, ..., 4 }weobtainthatm=n=k=r=0,i.e.thepolynomialF2 istrivialand beyond our consideration.
Thus, we proved the relation (i) =⇒ (ii) in Theorem 1.3. The relation (ii) =⇒ (iii) can be directly checked. Relation (iii) =⇒ (iv) follows from The- orem 2.2, and the relation (iv) =⇒ (i) is evident.
6. Polynomials of higher degrees
We start with the proof of Theorem 1.4. For this purpose we consider two
examples.
Example 6.1. Let n = 2k, k ∈ N. Then the (non-trivial) polynomial F (A,B):=
(AB)k of degree n is admissible.
Proof. For k = 1 this statement is well known (see, for example, equality
(2.3)). For k > 1 it follows from the following statement:
Proposition 6.2. Let F(A,B) be an admissible polynomial, then F(A,B)k is admissible for each k ∈ N.