In [1]:

    

import sympy
sympy.init_printing()


    

In [103]:

    

M, J, Q = sympy.symbols('M, J, Q', positive=True)
t, r = sympy.symbols('t, r', positive=True)
th, ph = sympy.symbols('theta, phi')

coords = t, r, th, ph


Ph = sympy.symbols('Phi', positive=True)
u, v, ut, al = sympy.symbols('u, v, u^t, alpha')
bx, by = sympy.symbols('beta^x, beta^y')

ud, vd = sympy.symbols('u_i, v_i')

vsq = u**2 + v**2
#W = al*ut
W = 1 / sympy.sqrt(1 - ud*u + vd*v)


    

In [4]:

    

w = Ph, u, v, ut
U = Ph*W, Ph*W**2*u, Ph*W**2*v, Ph*W**2*al*(ud * (u - bx/al) + vd * (v - by/al))
F_u = Ph*W*(u - bx/al), Ph*W**2*ud*(u - bx/al) + Ph, Ph*W**2*vd*(u - bx/al), Ph*W**2*(u - bx/al)*(ud*bx + vd*by - al)


    

In [22]:

    

for x in U:
    print(x.diff(Ph))


    

    




1/sqrt(-u*u_i + v*v_i + 1)
u/(-u*u_i + v*v_i + 1)
v/(-u*u_i + v*v_i + 1)
alpha*(u_i*(u - beta^x/alpha) + v_i*(v - beta^y/alpha))/(-u*u_i + v*v_i + 1)

In [23]:

    

for x in U:
    print(x.diff(u))


    

    




Phi*u_i/(2*(-u*u_i + v*v_i + 1)**(3/2))
Phi*u*u_i/(-u*u_i + v*v_i + 1)**2 + Phi/(-u*u_i + v*v_i + 1)
Phi*u_i*v/(-u*u_i + v*v_i + 1)**2
Phi*alpha*u_i*(u_i*(u - beta^x/alpha) + v_i*(v - beta^y/alpha))/(-u*u_i + v*v_i + 1)**2 + Phi*alpha*u_i/(-u*u_i + v*v_i + 1)

In [24]:

    

for x in U:
    print(x.diff(v))


    

    




-Phi*v_i/(2*(-u*u_i + v*v_i + 1)**(3/2))
-Phi*u*v_i/(-u*u_i + v*v_i + 1)**2
-Phi*v*v_i/(-u*u_i + v*v_i + 1)**2 + Phi/(-u*u_i + v*v_i + 1)
-Phi*alpha*v_i*(u_i*(u - beta^x/alpha) + v_i*(v - beta^y/alpha))/(-u*u_i + v*v_i + 1)**2 + Phi*alpha*v_i/(-u*u_i + v*v_i + 1)

In [29]:

    

for x in U:
    print(x.diff(ut))


    

    




Phi*alpha
2*Phi*alpha**2*u*u^t
2*Phi*alpha**2*u^t*v
2*Phi*alpha**3*u^t*(u_i*(u - beta^x/alpha) + v_i*(v - beta^y/alpha))

In [30]:

    

for x in F_u:
    print(x.diff(Ph))


    

    




alpha*u^t*(u - beta^x/alpha)
alpha**2*u^t**2*u_i*(u - beta^x/alpha) + 1
alpha**2*u^t**2*v_i*(u - beta^x/alpha)
alpha**2*u^t**2*(u - beta^x/alpha)*(-alpha + beta^x*u_i + beta^y*v_i)

In [31]:

    

for x in F_u:
    print(x.diff(ut))


    

    




Phi*alpha*(u - beta^x/alpha)
2*Phi*alpha**2*u^t*u_i*(u - beta^x/alpha)
2*Phi*alpha**2*u^t*v_i*(u - beta^x/alpha)
2*Phi*alpha**2*u^t*(u - beta^x/alpha)*(-alpha + beta^x*u_i + beta^y*v_i)

In [34]:

    

for x in F_u:
    print(x.diff(u))


    

    




Phi*u_i*(u - beta^x/alpha)/(2*(-u*u_i + v*v_i + 1)**(3/2)) + Phi/sqrt(-u*u_i + v*v_i + 1)
Phi*u_i**2*(u - beta^x/alpha)/(-u*u_i + v*v_i + 1)**2 + Phi*u_i/(-u*u_i + v*v_i + 1)
Phi*u_i*v_i*(u - beta^x/alpha)/(-u*u_i + v*v_i + 1)**2 + Phi*v_i/(-u*u_i + v*v_i + 1)
Phi*u_i*(u - beta^x/alpha)*(-alpha + beta^x*u_i + beta^y*v_i)/(-u*u_i + v*v_i + 1)**2 + Phi*(-alpha + beta^x*u_i + beta^y*v_i)/(-u*u_i + v*v_i + 1)

In [35]:

    

for x in F_u:
    print(x.diff(v))


    

    




-Phi*v_i*(u - beta^x/alpha)/(2*(-u*u_i + v*v_i + 1)**(3/2))
-Phi*u_i*v_i*(u - beta^x/alpha)/(-u*u_i + v*v_i + 1)**2
-Phi*v_i**2*(u - beta^x/alpha)/(-u*u_i + v*v_i + 1)**2
-Phi*v_i*(u - beta^x/alpha)*(-alpha + beta^x*u_i + beta^y*v_i)/(-u*u_i + v*v_i + 1)**2

In [67]:

    

ud = u
bx = 0
W = 1 / sympy.sqrt(1 - ud*u)
dudw = sympy.zeros(3,3)
dudw[0,0] = W
dudw[0,1] = Ph*ud*W**3
dudw[0,2] = Ph*al 

dudw[1,0] = W**2*ud
dudw[1,1] = Ph*W**2*(1 + 2*ud*ud*W**2)
dudw[1,2] = 2*Ph*al*W*ud

dudw[2,0] = W**2*al*ud*(u-bx/al)
dudw[2,1] = Ph*W**2*al*((1 + 2*W**2*ud*ud)*(u-bx/al) + ud)
dudw[2,2] = 2*Ph*al**2*W*ud*(u-bx/al)


    

In [68]:

    

dudw


    

    
Out[68]:





$$\left[\begin{matrix}\frac{1}{\sqrt{- u^{2} + 1}} & \frac{\Phi u}{\left(- u^{2} + 1\right)^{\frac{3}{2}}} & \Phi \alpha\\\frac{u}{- u^{2} + 1} & \frac{\Phi \left(\frac{2 u^{2}}{- u^{2} + 1} + 1\right)}{- u^{2} + 1} & \frac{2 \Phi \alpha u}{\sqrt{- u^{2} + 1}}\\\frac{\alpha u^{2}}{- u^{2} + 1} & \frac{\Phi \alpha}{- u^{2} + 1} \left(u \left(\frac{2 u^{2}}{- u^{2} + 1} + 1\right) + u\right) & \frac{2 \Phi \alpha^{2} u^{2}}{\sqrt{- u^{2} + 1}}\end{matrix}\right]$$

In [69]:

    

dudw_inv = dudw.inv()
dudw_inv.simplify()


    

In [70]:

    

dudw_inv


    

    
Out[70]:





$$\left[\begin{matrix}2 \sqrt{- u^{2} + 1} & 2 u - \frac{2}{u} & \frac{- u^{2} + 1}{\alpha u^{2}}\\0 & \frac{1}{\Phi} \left(u^{2} - 1\right) & \frac{- u^{2} + 1}{\Phi \alpha u}\\- \frac{1}{\Phi \alpha} & \frac{- u^{2} + 2}{\Phi \alpha u \sqrt{- u^{2} + 1}} & - \frac{1}{\Phi \alpha^{2} u^{2} \sqrt{- u^{2} + 1}}\end{matrix}\right]$$

In [71]:

    

dfdw = sympy.zeros(3,3)

dfdw[0,0] = W*(u-bx/al)
dfdw[0,1] = Ph*W*(1 + W**2*ud*(u-bx/al))
dfdw[0,2] = Ph*al*(u-bx/al)

dfdw[1,0] = W**2*ud*(u-bx/al) + 1
dfdw[1,1] = Ph*W**2*((1 + 2*W**2*ud*u)*(u-bx/al) + u)
dfdw[1,2] = 2*Ph*al*W*ud *(u-bx/al)

dfdw[2,0] = W**2*(u-bx/al)*(ud*bx - al)
dfdw[2,1] = Ph*W**2*((u-bx/al)*(bx + 2*W**2*u*(ud*bx-al)) + (ud*bx-al))
dfdw[2,2] = 2*Ph*al*W*(u-bx/al)*(ud*bx - al)

dfdw


    

    
Out[71]:





$$\left[\begin{matrix}\frac{u}{\sqrt{- u^{2} + 1}} & \frac{\Phi \left(\frac{u^{2}}{- u^{2} + 1} + 1\right)}{\sqrt{- u^{2} + 1}} & \Phi \alpha u\\\frac{u^{2}}{- u^{2} + 1} + 1 & \frac{\Phi}{- u^{2} + 1} \left(u \left(\frac{2 u^{2}}{- u^{2} + 1} + 1\right) + u\right) & \frac{2 \Phi \alpha u^{2}}{\sqrt{- u^{2} + 1}}\\- \frac{\alpha u}{- u^{2} + 1} & \frac{\Phi}{- u^{2} + 1} \left(- \frac{2 \alpha u^{2}}{- u^{2} + 1} - \alpha\right) & - \frac{2 \Phi \alpha^{2} u}{\sqrt{- u^{2} + 1}}\end{matrix}\right]$$

In [72]:

    

A = dfdw*dudw_inv


    

In [73]:

    

A.simplify()


    

In [75]:

    

evals = A.eigenvals()


    

In [81]:

    

P, D = A.diagonalize()


    

In [79]:

    

print(evals)


    

    




{-((-3*u**2 + 2)**2/u**2 - 3/u**2)/(3*(-1/2 + sqrt(3)*I/2)*(sqrt(-4*((-3*u**2 + 2)**2/u**2 - 3/u**2)**3 + (27*(-2*u**2 + 1)/u + 2*(-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/u**3)**2)/2 + 27*(-2*u**2 + 1)/(2*u) + (-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/(2*u**3))**(1/3)) - (-1/2 + sqrt(3)*I/2)*(sqrt(-4*((-3*u**2 + 2)**2/u**2 - 3/u**2)**3 + (27*(-2*u**2 + 1)/u + 2*(-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/u**3)**2)/2 + 27*(-2*u**2 + 1)/(2*u) + (-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/(2*u**3))**(1/3)/3 - (-3*u**2 + 2)/(3*u): 1, -((-3*u**2 + 2)**2/u**2 - 3/u**2)/(3*(sqrt(-4*((-3*u**2 + 2)**2/u**2 - 3/u**2)**3 + (27*(-2*u**2 + 1)/u + 2*(-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/u**3)**2)/2 + 27*(-2*u**2 + 1)/(2*u) + (-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/(2*u**3))**(1/3)) - (sqrt(-4*((-3*u**2 + 2)**2/u**2 - 3/u**2)**3 + (27*(-2*u**2 + 1)/u + 2*(-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/u**3)**2)/2 + 27*(-2*u**2 + 1)/(2*u) + (-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/(2*u**3))**(1/3)/3 - (-3*u**2 + 2)/(3*u): 1, -((-3*u**2 + 2)**2/u**2 - 3/u**2)/(3*(-1/2 - sqrt(3)*I/2)*(sqrt(-4*((-3*u**2 + 2)**2/u**2 - 3/u**2)**3 + (27*(-2*u**2 + 1)/u + 2*(-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/u**3)**2)/2 + 27*(-2*u**2 + 1)/(2*u) + (-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/(2*u**3))**(1/3)) - (-1/2 - sqrt(3)*I/2)*(sqrt(-4*((-3*u**2 + 2)**2/u**2 - 3/u**2)**3 + (27*(-2*u**2 + 1)/u + 2*(-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/u**3)**2)/2 + 27*(-2*u**2 + 1)/(2*u) + (-3*u**2 + 2)**3/u**3 - 9*(-3*u**2 + 2)/(2*u**3))**(1/3)/3 - (-3*u**2 + 2)/(3*u): 1}

In [80]:

    

for x in evals:
    print(x.simplify())


    

    




(2**(2/3)*u**2*((-54*u**4 + u**3*sqrt((-4*((-3*u**2 + 2)**2 - 3)**3 + (27*u**2*(-2*u**2 + 1) + 27*u**2 + 2*(-3*u**2 + 2)**3 - 18)**2)/u**6) + 54*u**2 - 2*(3*u**2 - 2)**3 - 18)/u**3)**(2/3)*(1 - sqrt(3)*I)**2 + 4*u*((u**3*sqrt((-4*((-3*u**2 + 2)**2 - 3)**3 + (27*u**2*(-2*u**2 + 1) + 27*u**2 + 2*(-3*u**2 + 2)**3 - 18)**2)/u**6) + 27*u**2*(-2*u**2 + 1) + 27*u**2 - 2*(3*u**2 - 2)**3 - 18)/u**3)**(1/3)*(1 - sqrt(3)*I)*(3*u**2 - 2) + 8*2**(1/3)*((3*u**2 - 2)**2 - 3))/(12*u**2*((u**3*sqrt((-4*((-3*u**2 + 2)**2 - 3)**3 + (27*u**2*(-2*u**2 + 1) + 27*u**2 + 2*(-3*u**2 + 2)**3 - 18)**2)/u**6) + 27*u**2*(-2*u**2 + 1) + 27*u**2 - 2*(3*u**2 - 2)**3 - 18)/u**3)**(1/3)*(1 - sqrt(3)*I))
-3*u**2/(-27*u**3 + 27*u + 3*sqrt(6)*sqrt(27*u**4 - 54*u**2 + 36 - 8/u**2 + u**(-4)) - 9/u - 1/u**3)**(1/3) + u - (-27*u**3 + 27*u + 3*sqrt(6)*sqrt(27*u**4 - 54*u**2 + 36 - 8/u**2 + u**(-4)) - 9/u - 1/u**3)**(1/3)/3 + 4/(-27*u**3 + 27*u + 3*sqrt(6)*sqrt(27*u**4 - 54*u**2 + 36 - 8/u**2 + u**(-4)) - 9/u - 1/u**3)**(1/3) - 2/(3*u) - 1/(3*u**2*(-27*u**3 + 27*u + 3*sqrt(6)*sqrt(27*u**4 - 54*u**2 + 36 - 8/u**2 + u**(-4)) - 9/u - 1/u**3)**(1/3))
(2**(2/3)*u**2*((-54*u**4 + u**3*sqrt((-4*((-3*u**2 + 2)**2 - 3)**3 + (27*u**2*(-2*u**2 + 1) + 27*u**2 + 2*(-3*u**2 + 2)**3 - 18)**2)/u**6) + 54*u**2 - 2*(3*u**2 - 2)**3 - 18)/u**3)**(2/3)*(1 + sqrt(3)*I)**2 + 4*u*((u**3*sqrt((-4*((-3*u**2 + 2)**2 - 3)**3 + (27*u**2*(-2*u**2 + 1) + 27*u**2 + 2*(-3*u**2 + 2)**3 - 18)**2)/u**6) + 27*u**2*(-2*u**2 + 1) + 27*u**2 - 2*(3*u**2 - 2)**3 - 18)/u**3)**(1/3)*(1 + sqrt(3)*I)*(3*u**2 - 2) + 8*2**(1/3)*((3*u**2 - 2)**2 - 3))/(12*u**2*((u**3*sqrt((-4*((-3*u**2 + 2)**2 - 3)**3 + (27*u**2*(-2*u**2 + 1) + 27*u**2 + 2*(-3*u**2 + 2)**3 - 18)**2)/u**6) + 27*u**2*(-2*u**2 + 1) + 27*u**2 - 2*(3*u**2 - 2)**3 - 18)/u**3)**(1/3)*(1 + sqrt(3)*I))

In [74]:

    

A


    

    
Out[74]:





$$\left[\begin{matrix}u & - \sqrt{- u^{2} + 1} & \frac{\sqrt{- u^{2} + 1}}{\alpha u}\\2 \sqrt{- u^{2} + 1} & 2 u - \frac{2}{u} & \frac{1}{\alpha u^{2}}\\0 & - \alpha & 0\end{matrix}\right]$$

In [66]:

    

dudw.det()


    

    
Out[66]:





$$\frac{2 \Phi^{2} \alpha^{2} u^{4}}{- u^{6} + 3 u^{4} - 3 u^{2} + 1} + \frac{3 \Phi^{2} \alpha^{2} u^{2}}{u^{4} - 2 u^{2} + 1} - \frac{2 \Phi^{2} \alpha u^{4}}{- u^{6} + 3 u^{4} - 3 u^{2} + 1} - \frac{4 \Phi^{2} \alpha u^{2}}{u^{4} - 2 u^{2} + 1}$$

In [78]:

    

evals.simplify()


    

    




---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-78-1eaca1d5ce9f> in <module>()
----> 1 evals.simplify()

AttributeError: 'dict' object has no attribute 'simplify'

In [90]:

    

B = sympy.zeros(3,3)
B[0,0] = u
B[0,1] = -1/W
B[0,2] = 1/(al*u*W)
B[1,0] = 2/W
B[1,1] = 2*u - 2/u
B[1,2] = 1/(al*u**2)
B[2,1] = -al
B


    

    
Out[90]:





$$\left[\begin{matrix}u & - \sqrt{- u^{2} + 1} & \frac{\sqrt{- u^{2} + 1}}{\alpha u}\\2 \sqrt{- u^{2} + 1} & 2 u - \frac{2}{u} & \frac{1}{\alpha u^{2}}\\0 & - \alpha & 0\end{matrix}\right]$$

In [97]:

    

w = sympy.symbols('w')
B = B.subs(W,w)


    

In [98]:

    

PB, DB = B.diagonalize()


    

In [105]:

    

dudw = sympy.zeros(3,3)
dudw[0,0] = w
dudw[0,1] = Ph*ud*w**3
dudw[0,2] = Ph*vd*w**3

dudw[1,0] = w**2*ud
dudw[1,1] = Ph*w**2*(1+2*ud*ud*w**2)
dudw[1,2] = Ph*w**2*2*ud*vd*w**2

dudw[2,0] = w**2*vd
dudw[2,1] = Ph*w**2*2*ud*vd*w**2
dudw[2,2] = Ph*w**2*(1+2*vd**2*w**2)

dudw


    

    
Out[105]:





$$\left[\begin{matrix}w & \Phi u_{i} w^{3} & \Phi v_{i} w^{3}\\u_{i} w^{2} & \Phi w^{2} \left(2 u_{i}^{2} w^{2} + 1\right) & 2 \Phi u_{i} v_{i} w^{4}\\v_{i} w^{2} & 2 \Phi u_{i} v_{i} w^{4} & \Phi w^{2} \left(2 v_{i}^{2} w^{2} + 1\right)\end{matrix}\right]$$

In [106]:

    

dudw.det()


    

    
Out[106]:





$$\Phi^{2} u_{i}^{2} w^{7} + \Phi^{2} v_{i}^{2} w^{7} + \Phi^{2} w^{5}$$

In [109]:

    

dudw_inv = dudw.inv()


    

In [125]:

    

# set bi = 0
dfdw = sympy.zeros(3,3)

dfdw[0,0] = w*(u-bx)
dfdw[0,1] = Ph*w*(1+w**2*ud*(u-bx))
dfdw[0,2] = Ph*w*w**2*vd*(u-bx)

dfdw[1,0] = w**2*ud*(u-bx) + 1
dfdw[1,1] = Ph*w**2*((u-bx)*(1+2*w**2*ud*u)+u)
dfdw[1,2] = Ph*w**2*((u-bx)*(vd*u*w**2))

dfdw[2,0] = w**2*vd*(u-bx)
dfdw[2,1] = Ph*w**2*((u-bx)*(2*w**2*ud*v))
dfdw[2,2] = Ph*w**2*((u-bx)*(1+2*w**2*vd*v)+u)

dfdw


    

    
Out[125]:





$$\left[\begin{matrix}w \left(- \beta^{x} + u\right) & \Phi w \left(u_{i} w^{2} \left(- \beta^{x} + u\right) + 1\right) & \Phi v_{i} w^{3} \left(- \beta^{x} + u\right)\\u_{i} w^{2} \left(- \beta^{x} + u\right) + 1 & \Phi w^{2} \left(u + \left(- \beta^{x} + u\right) \left(2 u u_{i} w^{2} + 1\right)\right) & \Phi u v_{i} w^{4} \left(- \beta^{x} + u\right)\\v_{i} w^{2} \left(- \beta^{x} + u\right) & 2 \Phi u_{i} v w^{4} \left(- \beta^{x} + u\right) & \Phi w^{2} \left(u + \left(- \beta^{x} + u\right) \left(2 v v_{i} w^{2} + 1\right)\right)\end{matrix}\right]$$

In [126]:

    

a = dfdw * dudw_inv


    

In [127]:

    

a.simplify()


    

In [128]:

    

a


    

    
Out[128]:





$$\left[\begin{matrix}\frac{1}{u_{i}^{2} w^{2} + v_{i}^{2} w^{2} + 1} \left(- \beta^{x} u_{i}^{2} w^{2} - \beta^{x} v_{i}^{2} w^{2} - \beta^{x} + u u_{i}^{2} w^{2} + u v_{i}^{2} w^{2} + u - u_{i}\right) & \frac{v_{i}^{2} w^{2} + 1}{w \left(u_{i}^{2} w^{2} + v_{i}^{2} w^{2} + 1\right)} & - \frac{u_{i} v_{i} w}{u_{i}^{2} w^{2} + v_{i}^{2} w^{2} + 1}\\\frac{1}{w \left(u_{i}^{2} w^{2} + v_{i}^{2} w^{2} + 1\right)} \left(2 \beta^{x} u u_{i}^{2} w^{4} + \beta^{x} u v_{i}^{2} w^{4} - 2 \beta^{x} u_{i}^{3} w^{4} - 2 \beta^{x} u_{i} v_{i}^{2} w^{4} - 2 u^{2} u_{i}^{2} w^{4} - u^{2} v_{i}^{2} w^{4} + 2 u u_{i}^{3} w^{4} + 2 u u_{i} v_{i}^{2} w^{4} - u u_{i} w^{2} + 2 u_{i}^{2} w^{2} + 2 v_{i}^{2} w^{2} + 1\right) & \frac{1}{u_{i}^{2} w^{2} + v_{i}^{2} w^{2} + 1} \left(- \beta^{x} u u_{i} v_{i}^{2} w^{4} - 2 \beta^{x} u u_{i} w^{2} + \beta^{x} u_{i}^{2} w^{2} - \beta^{x} v_{i}^{2} w^{2} - \beta^{x} + u^{2} u_{i} v_{i}^{2} w^{4} + 2 u^{2} u_{i} w^{2} - u u_{i}^{2} w^{2} + 2 u v_{i}^{2} w^{2} + 2 u - u_{i}\right) & \frac{v_{i}}{u_{i}^{2} w^{2} + v_{i}^{2} w^{2} + 1} \left(u_{i} w^{2} \left(\beta^{x} - u\right) - w^{2} \left(u \left(\beta^{x} - u\right) \left(u_{i}^{2} w^{2} + 1\right) + u_{i} \left(u - \left(\beta^{x} - u\right) \left(2 u u_{i} w^{2} + 1\right)\right)\right) - 1\right)\\- \frac{w}{u_{i}^{2} w^{2} + v_{i}^{2} w^{2} + 1} \left(- 2 \beta^{x} u_{i}^{2} v w^{2} + 2 \beta^{x} u_{i}^{2} v_{i} w^{2} - 2 \beta^{x} v v_{i}^{2} w^{2} + 2 \beta^{x} v_{i}^{3} w^{2} + 2 u u_{i}^{2} v w^{2} - 2 u u_{i}^{2} v_{i} w^{2} + 2 u v v_{i}^{2} w^{2} - 2 u v_{i}^{3} w^{2} + u v_{i}\right) & \frac{u_{i} w^{2}}{u_{i}^{2} w^{2} + v_{i}^{2} w^{2} + 1} \left(- 2 \beta^{x} v + 2 \beta^{x} v_{i} + 2 u v - 3 u v_{i}\right) & \frac{1}{u_{i}^{2} w^{2} + v_{i}^{2} w^{2} + 1} \left(2 u_{i}^{2} v v_{i} w^{4} \left(\beta^{x} - u\right) + v_{i}^{2} w^{2} \left(\beta^{x} - u\right) + \left(u - \left(\beta^{x} - u\right) \left(2 v v_{i} w^{2} + 1\right)\right) \left(u_{i}^{2} w^{2} + 1\right)\right)\end{matrix}\right]$$

In [129]:

    

evals = a.eigenvals()


    

In [ ]:

    

for x in evals:
    print(x.simplify())


    

In [ ]: