f=ρ0*i0*(-1/(i0*#5)^(γ/(γ-1)))*Exp[(-γ+#1-(#2^2+#3^2+#4^2)/(2*#5))/(γ-1)]&

expre={-V5,V2,V3,V4,1-(V2^2+V3^2+V4^2)/(2*V5)}
MatrixForm[{-V5,V2,V3,V4,1-(V2^2+V3^2+V4^2)/(2*V5)}]

D1={{D[f[V1,V2,V3,V4,V5],V1]},{ D[f[V1,V2,V3,V4,V5],V2]},{ D[f[V1,V2,V3,V4,V5],V3]},{ D[f[V1,V2,V3,V4,V5],V4]},{ D[f[V1,V2,V3,V4,V5],V5]}}

MatrixForm[FullSimplify[D1]]

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]

cc =((E^((V1 - (V2^2 + V3^2 + V4^2)/(2*V5) - γ)/
        (-1 + γ))*i0*ρ0)/
     ((i0*V5)^(γ/(-1 + γ))*(-1 + γ)))

A1=FullSimplify[{{ee *V5,-ee *V2,-ee *V3,-ee *V4,-ee *(1-(V2^2 +V3^2+V4^2)/(2V5))},{-ee *V2,ee *V2^2/V5+f[V1,V2,V3,V4,V5],ee *V3*V2/V5,ee *V4*V2/V5,-ee*V2*(V2^2 +V3^2+V4^2-2*γ*V5^2)/(2*V5^2)},{-ee *V3,ee *V2*V3/V5,ee *V3^2/V5+f[V1,V2,V3,V4,V5],ee *V4*V3/V5,-ee*V3*(V2^2 +V3^2+V4^2-2*γ*V5^2)/(2*V5^2)},{-ee *V4,ee *V2*V4/V5,ee *V3*V4/V5,f[V1,V2,V3,V4,V5]+ee *V4^2/V5,-ee*V4*(V2^2 +V3^2+V4^2-2*γ*V5^2)/(2*V5^2)},{-ee *(1-(V2^2 +V3^2+V4^2)/(2V5)),-ee*V2*F,-ee*V3*F,-ee*V4*F/(2*V5^2),f[V1,V2,V3,V4,V5]*F-ee*(1-(V2^2 +V3^2+V4^2)/(2V5))*F}}]