Re: bound on eigenvalues...

Consider the a symmetric 3 x 3 matrix

A=
a_{11} a_{12} a_{13}
a_{12} a_{22} a_{23}
a_{13} a_{23} a_{33} ,
and denote

D:=(a_{11} -a_{22})^2 +(a_{11}-a_{33})^2 +(a_{22} -a_{33} )^2 +

+ 6*( a_{12}^2 +a_{13}^2 +a_{23}^2) ,

M:= (a_{11} +a_{22} +a_{33} )/3 .

Proposition. If r_3 =< r_2 =< r_1 are the eigenvalues of A,
then :

i)

M+ (1/3)*sqrt(D*0.5) =< r_1 =< M+ (1/3)*sqrt(2*D)

M - (1/3)*sqrt(D*0.5) =< r_2 =< M +(1/3)*sqrt(D*0.5)

M-(1/3)*sqrt(2*D) =< r_3 =< M - (1/3)*sqrt(D*0.5) .
...............................
ii)
sqrt(D*0.5) =< r_1 - r_3 =< sqrt(2*D/3) .
...............................
iii)

min_{1=<i<k=< 3} (x_i - x_k) =< (1/3)*sqrt(D*1.5) .
..........................
Note: sqrt(A) := A^{1/2}
..........................
In your case D=0.35746806 , M= 0.167 .
For instance you find

0.30792315873... =< r_1 =< 0.3685021126423...

.