F[U1_,U2_,U3_,U4_,U5_]:= {U2,U2^2/U1 + (γ-1)*(U5-((U2^2)+(U3^2)+(U4^2))/(U1*2)),U2*U3/U1,U2*U4/U1,U2/U1*U5+(γ-1)*(U5-((U2^2)+(U3^2)+(U4^2))/(U1*2))*U2/U1}

A={ D[F[U1,U2,U3,U4,U5],U1], D[F[U1,U2,U3,U4,U5],U2],  D[F[U1,U2,U3,U4,U5],U3],D[F[U1,U2,U3,U4,U5],U4],D[F[U1,U2,U3,U4,U5],U5]}

MatrixForm[Simplify[Transpose[A]]]

FullSimplify[Eigenvalues[Transpose[A]]]

Aclassic = {{0,1,0},{-u^2+(γ-1)/2*u^2,2*u - (γ-1)*u,(γ-1)},{u^3*(γ-1)-γ*e*u,-(γ-1)*3*u^2/2 + γ*e,γ*u}}

MatrixForm[Aclassic]

FullSimplify[Eigenvalues[Aclassic]]

FullSimplify[Eigenvectors[Aclassic]]

g=(#1/#2^γ)&
FullSimplify[g[p1,ρ1]^(α*ρ1)*g[p2,ρ2]^((α-1)*ρ2)/g[α*p1+(α-1)*p2,α*ρ1+(α-1)*ρ2]^(α*ρ1+(α-1)*ρ2)]