For the nucleotide substitution model of Section 10.5, show that has eigenvalues 0, (+++), (+ ++),

For the nucleotide substitution model of Section 10.5, show that Λ

has eigenvalues 0, −(γ+λ+δ+κ), −(α++γ+λ), and −(δ+κ+β+σ)

and corresponding right eigenvectors 1 =

1, 1, 1, 1 t

u =

1, 1, −c5 c2

, −c5 c2 t

v =

α(c5 − c3) + κc2

δ(c3 − c2) − c5

,

−(c5 − c3) − δc2

δ(c3 − c2) − c5

, 1, 1 t

w =

1, 1,

β(c2 − c6) + λc5

γ(c6 − c5) − σc2

,

−σ(c2 − c6) − γc5

γ(c6 − c5) − σc2 respectively, where the constants c1,...,c6 are same ones defined in equations (10.12) through (10.15).