According to this theorem, the eigenvector of A belonging to eigenvalue 3 is orthogonal to p 1, so it is the solution of the system of equations x 1+x2+x3=0. A set of basic solution systems of the equation p2=( 1,-1, 0)', p3=( 1, 1, -2)' is the eigenvector of A belonging to eigenvalue 3 (here, p2 and p3 are appropriately selected to make them equal to p 1.

Let p 1, p2, p3 Huasong unit vector be P=(p 1/√3, p2/√2, p3/√6), then P is an orthogonal matrix, AP = pλ, where λ = Diag (6 6,3,3), so a = pλ p'

4 1 1

1 4 1

1 1 4

If the previous p2 and p3 are a set of basic solution systems randomly selected, then when using P = (P 1, P2, P3) to find A according to AP = pλ, it is necessary to calculate the inverse matrix of P, and the calculation amount is slightly larger.