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Factoring a symbolic polynomial

 llambda, mu = var('λ, μ')
uVars    = list(var(', '.join([f'u{n}' for n in range(1, 3 + 1)])))
aVars    = list(var(', '.join([f'a{n}' for n in range(1, 3 + 1)])))

U = Matrix([ [0, -uVars[2], uVars[1]], [uVars[2], 0, -uVars[0]], [-uVars[1], uVars[0], 0] ])
a = Matrix([ [aVars[0], 0, 0], [0, aVars[1], 0], [0, 0, aVars[2]] ])
I = matrix.identity(3)

L = a*llambda + U

# Characteristic polynomial

charPoly = det((L - mu*I))
factor(charPoly)

The above code computes a characteristic polynomial and yields:

a1a2a3λ^3 - a1a2λ^2μ - a1a3λ^2μ - a2a3λ^2μ + a1u1^2λ + a2u2^2λ + a3u3^2λ + a1λμ^2 + a2λμ^2 + a3λμ^2 - u1^2μ - u2^2μ - u3^2*μ - μ^3

However, this is not the simplification I desire. I want this: $$(a_1\lambda - \mu)(a_2\lambda - \mu)(a_3\lambda - \mu) - (u_1^2 + u_2^2 + u_3^2)\mu + (a_1 \mu_1^2 + a_2 u_2^2 + a_3 u_3^2)\lambda$$

Is there a way to obtain that sort of factorization?

Factoring a symbolic polynomial

llambda, mu = var('λ, μ') uVars = list(var(', '.join([f'u{n}' for n in range(1, 3 + 1)]))) aVars = list(var(', '.join([f'a{n}' for n in range(1, 3 + 1)]))) 1)]))) U = Matrix([ [0, -uVars[2], uVars[1]], [uVars[2], 0, -uVars[0]], [-uVars[1], uVars[0], 0] ]) a = Matrix([ [aVars[0], 0, 0], [0, aVars[1], 0], [0, 0, aVars[2]] ]) I = matrix.identity(3) L = a*llambda + U # Characteristic polynomial charPoly = det((L - mu*I)) factor(charPoly) The above code computes a characteristic polynomial and yields: a1a2a3λ^3 - a1a2λ^2μ - a1a3λ^2μ - a2a3λ^2μ + a1u1^2λ + a2u2^2λ + a3u3^2λ + a1λμ^2 + a2λμ^2 + a3λμ^2 - u1^2μ - u2^2μ a1*a2*a3*λ^3 - a1*a2*λ^2*μ - a1*a3*λ^2*μ - a2*a3*λ^2*μ + a1*u1^2*λ + a2*u2^2*λ + a3*u3^2*λ + a1*λ*μ^2 + a2*λ*μ^2 + a3*λ*μ^2 - u1^2*μ - u2^2*μ - u3^2*μ - μ^3μ^3 However, this is not the simplification I desire. I want this: this: $$(a_1\lambda - \mu)(a_2\lambda - \mu)(a_3\lambda - \mu) \mu) - (u_1^2 + u_2^2 + u_3^2)\mu + (a_1 \mu_1^2 + a_2 u_2^2 u_2^2 + a_3 u_3^2)\lambda$$ Is there a way to obtain that sort of factorization?

          3    None 
      
    
        
        updated 2020-07-03 00:27:06 +0100  
    slelievre 
 17684  ●22 ●160 ●349  
        http://carva.org/samue...
         
 
 
 
  
 
 
 
  Factoring a symbolic polynomial
 The following code:
 llambda, mu = var('λ, μ')
 uVars  = list(var(', '.join([f'u{n}' for n in range(1, 3 + 1)])))
 aVars  = list(var(', '.join([f'a{n}' for n in range(1, 3 + 1)])))
 1)])))

U = Matrix([ [0, Matrix([[0, -uVars[2], uVars[1]], uVars[1]],
            [uVars[2], 0, -uVars[0]], -uVars[0]],
            [-uVars[1], uVars[0], 0] ])
a = Matrix([ [aVars[0], 0]])
a = Matrix([[aVars[0], 0, 0], 0],
            [0, aVars[1], 0], 0],
            [0, 0, aVars[2]] ])
aVars[2]]])
I = matrix.identity(3)

L = a*llambda + U

# Characteristic polynomial

charPoly = det((L - mu*I))
factor(charPoly)
 The above code computes a characteristic polynomial and yields: 
 a1*a2*a3*λ^3 - a1*a2*λ^2*μ - a1*a3*λ^2*μ - a2*a3*λ^2*μ
+ a1*u1^2*λ + a2*u2^2*λ + a3*u3^2*λ + a1*λ*μ^2
+ a2*λ*μ^2 + a3*λ*μ^2 - u1^2*μ - u2^2*μ - u3^2*μ - μ^3
 However, this is not the simplification I desire. I want this:
$$(a_1\lambda - \mu)(a_2\lambda - \mu)(a_3\lambda - \mu)
- (u_1^2 + u_2^2 + u_3^2)\mu + (a_1 \mu_1^2 + a_2 u_2^2
+ a_3 u_3^2)\lambda$$
 Is there a way to obtain that sort of factorization?