Q 5.8 (1)  Herimite p.101

aa1 = {{1, I, 4}, {-I, 3, 0}, {4, 0, 
            2}};
MatrixForm[aa1]

Conjugate[
        Transpose[
          aa1]]-aa1   (*check Hermitian definition*)

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

bs = Chop[Conjugate[ϕt] . ϕ];
MatrixForm[bs]
as = Chop[Conjugate[ϕt] . aa1 . ϕ]
MatrixForm[as]

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

Q 5.8 (2)  skew-Herimitian  p.101

aa2 = {{-I, 1, 2 + I}, {-1, 0, 3 I}, {-2 + I, 3 I, 
            I}};
MatrixForm[aa2]

-Conjugate[Transpose[aa2]]-aa2

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

bs = Chop[Conjugate[ϕt] . ϕ];
MatrixForm[bs]
as = 
        Chop[Conjugate[ϕt] . aa2 . ϕ];
MatrixForm[as]

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