ex 5.20  symmetric   p.93

aa1 = {{1, -1, 0}, {-1, 2, -1}, {0, -1, 
            3}};
MatrixForm[aa1]

{λ, ϕt} = Eigensystem[aa1];
λ
ϕ = Transpose[ϕt]; MatrixForm[ϕ]

bs = Simplify[ϕt . ϕ]
as = Simplify[ϕt . aa1 . ϕ]

Table[as[[i, i]]/bs[[i, i]], {i, 3}]
Simplify[λ/%]

example 5.21  skew-symmetric  p.94

aa2 = {{0, 9, -12}, {-9, 0, 20}, {12, -20, 
            0}};
MatrixForm[aa2]

{λ, ϕt} = Eigensystem[aa2];
λ
ϕ = Transpose[ϕt]; MatrixForm[ϕ]

bs = Simplify[Conjugate[ϕt] . ϕ]
as = Simplify[Conjugate[ϕt] . aa2 . ϕ]

Table[as[[i, i]]/bs[[i, i]], {i, 3}]

example 5.22  orthogonal   p.95

aa3 = {{2, 1, 2}, {-2, 2, 1}, {1, 2, -2}}/
          3;
MatrixForm[aa3]

Inverse[aa3] - Transpose[aa3]

{λ, ϕt} = Eigensystem[aa3];
λ
ϕ = Transpose[ϕt]; MatrixForm[ϕ]

bs = 
        Simplify[Conjugate[ϕt] . ϕ];
N[%]
as = 
        Simplify[
          Conjugate[ϕt] . aa3 . ϕ];
N[%]

Table[as[[i, i]]/bs[[i, i]], {i, 3}]

example 5.23  orthogonal   p.95

aa4 = {{16, 0, 0}, {48, -8, 0}, {84, -24, 
            4}};
MatrixForm[aa4]

{λ, ϕt} = Eigensystem[aa4];
λ
ϕ = Transpose[ϕt]; MatrixForm[ϕ]

case 1

bs = Inverse[ϕ] . ϕ
as = Inverse[ϕ] . aa4 . ϕ

Table[as[[i, i]]/bs[[i, i]], {i, 3}]

case 2

{λ1, ψt} = 
        Eigensystem[Transpose[aa4]];
λ1
ψ = Transpose[ψt]; MatrixForm[ψ]

bs = ψt . ϕ
as = ψt . aa4 . ϕ

Table[as[[i, i]]/bs[[i, i]], {i, 3}]