python - 在 Sympy 矩阵中查找复数的实部

Question

我从 sympy 的 MyMat.eigenvals().keys() 获取此矩阵(这是 4D 雅可比矩阵的特征值):

egV = [5728761814608399/400000000000000 + sqrt(116481370240190604554434649577057/100000000000000000000000000000 - 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3) - 542870217167939624258689632311630767295785372177/(20000000000000000000000000000000000000000000*sqrt(116481370240190604554434649577057/200000000000000000000000000000 + 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)) + 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3))) - 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)))/2 - sqrt(116481370240190604554434649577057/200000000000000000000000000000 + 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)) + 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3))/2, 5728761814608399/400000000000000 + sqrt(116481370240190604554434649577057/100000000000000000000000000000 - 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3) + 542870217167939624258689632311630767295785372177/(20000000000000000000000000000000000000000000*sqrt(116481370240190604554434649577057/200000000000000000000000000000 + 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)) + 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3))) - 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)))/2 + sqrt(116481370240190604554434649577057/200000000000000000000000000000 + 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)) + 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3))/2, 5728761814608399/400000000000000 - sqrt(116481370240190604554434649577057/200000000000000000000000000000 + 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)) + 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3))/2 - sqrt(116481370240190604554434649577057/100000000000000000000000000000 - 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3) - 542870217167939624258689632311630767295785372177/(20000000000000000000000000000000000000000000*sqrt(116481370240190604554434649577057/200000000000000000000000000000 + 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)) + 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3))) - 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)))/2, 5728761814608399/400000000000000 - sqrt(116481370240190604554434649577057/100000000000000000000000000000 - 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3) + 542870217167939624258689632311630767295785372177/(20000000000000000000000000000000000000000000*sqrt(116481370240190604554434649577057/200000000000000000000000000000 + 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)) + 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3))) - 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)))/2 + sqrt(116481370240190604554434649577057/200000000000000000000000000000 + 84525541314107370029729686916824478617779085658015330937307969/(20000000000000000000000000000000000000000000000000000000000*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3)) + 2*(704966930593670936740277648541223935870775995019826163729310179922474354087956414591800649953/8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 + sqrt(4176563721214961521026417694955136953714825217199604642869248664796023172422674692165308442655306961942463737618615469818663523407583102219569144299457956960070280847076949520807345)*I/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000)**(1/3))/2]

我只想获取实数部分，因此我尝试使用 re() :

outMat = [re(eg) for eg in egV]

但它返回此错误:

TypeError: cannot determine truth value of Relational

Answer 1

re 函数的文档字符串表示

Returns real part of expression. This function performs only elementary analysis and so it will fail to decompose properly more complicated expressions. If completely simplified result is needed then use Basic.as_real_imag() or perform complex expansion on instance of this function.

在 SymPy 1.3 中，egV.as_real_imag() 返回矩阵元组:实部和虚部。虚部全为零，这告诉我们特征值是实数。显然，I 的存在来自 casus irreducibilis三次方程的一个(方程有三个实根，但它们的代数公式必然涉及复数)。