1 b 1

b a 1

1 1 1

Similar diagonal matrix B=diag(0, 1, 4).

So tr(A)=2+a=tr(B)=5, and |A|=|B|=0.

So a=3.

So | a | =-(b- 1) 2

So b= 1.

So A=

1 1 1

1 3 1

1 1 1

And the eigenvalue of a is 0, 1, 4.

The basic system of AX=0 is A 1 = (1, 0,-1) t.

The basic system of (A-E)X is a2 = (1,-1,1) t.

The basic system of (A-4E)X is a3 = (1, 2,1) t.

unitized

a 1=( 1/√2,0,- 1/√2)^t

a2=( 1/√3,- 1/√3, 1/√3)^t

a3=( 1/√6,2/√6, 1/√6)^t

Let P=(a 1, a2, a3), then p is the expected orthogonal matrix.