Page 159 - Textos de Matemática Vol. 44

THE MAIN DIAGONAL PRODUCTS OF 3 × 3 NORMAL MATRICES 149
Using the polynomial
f(a11, a12, a21, a22)=(1−a11−a12−a21−a22 +a11a22 +a12a21)2−4a11a12a21a22, [11] and [5] showed that the set ∂Uni(3) in A(3) is definable as
∂Uni(3) = {A = (aij) ∈ D(3) : f(a11,a12,a21,a22) = 0}.
We shall consider the subset TU(3) of ∂Uni(3) consisting of matrices of the
form
a b c A = A(a,b,c) = c a b.
bca
So we have the relations a22 = a11 = a, a+b+c = 1 and f(a,b,c,a) = f (a, b, 1 − a − b, a) = 0. Explicitly this yields the expression xpr = a2 − 2a3 + a4 −2ab+2a3b+b2 +3a2b2 −2b3 +2ab3 +b4 = 0. From this by a system like Mathematica⃝c , you can easily extract b via Solve[xpr==0,b]. Factoring the expressions under the root obtained yields b as being defined locally as one of the following functions of a.
g1(a) = g2(a) = g3(a) = g4(a) =
1{1−a+ (1−√a)3 1+3√a}, 0≤a≤1; 2
1{1−a− (1−√a)3 1+3√a}, 0≤a≤1; 2
1{1−a+ (1+√a)3 1−3√a}, 0≤a≤1/9; 2
1{1−a− (1+√a)3 1−3√a}, 0≤a≤1/9. 2
These relatively simple expressions for solving an equation of fourth degree in b are something of a miracle.
For j=1,2and0≤a≤1 or j=3,4and0≤a≤1/9, wehave 0 ≤ a + gj(a) ≤ 1. This is not hard to check and we see that we have a real algebraic curve that has in an (a,b)-coordinate system the three vertices (0, 0), (1, 0), (0, 1). With ImplicitPlot[xpr==0,{a,0,1}] one gets a nice pic- ture.
Our considerations show that T U (3) is precisely the set of matrices A(a, b, c) for which 0 ≤ a ≤ 1, b is one of the gj(a) with a admissible, and c satisfies c = 1 − a − b.