Eigen: Functions of matrix argument

Eigen: Functions of matrix argument#

Matrix Exponential#

matA.Expm()#

Computes the matrix exponential of a square matrix \(A\), which is defined by the power series \(\displaystyle \exp(A) = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \ldots\)

See also: Wikipedia [1546], Wikipedia [1556].

See also: Eigen [216], Eigen [215].

Basic examples:

>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> expm(zeros(3))
[1.0  0.0  0.0]
[0.0  1.0  0.0]
[0.0  0.0  1.0]
>>> expm(eye(3))
[2.71828182845905               0.0               0.0]
[             0.0  2.71828182845905               0.0]
[             0.0               0.0  2.71828182845905]
>>> expm([[1,1,0],[1,0,1],[0,1,0]])
[ 3.86814500615414  2.26812870852145  0.841130841230196]
[ 2.26812870852145  2.44114713886289   1.42699786729125]
[0.841130841230196  1.42699786729125    1.6000162976327]
>>> expm([[1,1,0],[1,0,1],[0,1,0]], method='pade')
[ 3.86814500615414  2.26812870852145  0.841130841230196]
[ 2.26812870852145  2.44114713886289   1.42699786729125]
[0.841130841230196  1.42699786729125    1.6000162976327]
>>> expm([[1+j, 0], [1+j,1]])
[(1.46869393991589 + 2.28735528717884j)                        0.0]
[  (1.03776739863568 + 3.536943175722j)  (2.71828182845905 + 0.0j)]

Matrices with large entries are allowed:

>>> expm(matrix([[1,2],[2,3]])**25)
[5.65024064048415e+2050488462815550  9.14228140091932e+2050488462815550]
[9.14228140091932e+2050488462815550  1.47925220414035e+2050488462815551]

The identity \(\exp(A+B) = \exp(A) \exp(B)\) does not hold for noncommuting matrices:

>>> A = hilbert(3)
>>> B = A + eye(3)
>>> chop(mnorm(A*B - B*A))
0.0
>>> chop(mnorm(expm(A+B) - expm(A)*expm(B)))
0.0
>>> B = A + ones(3)
>>> mnorm(A*B - B*A)
1.8
>>> mnorm(expm(A+B) - expm(A)*expm(B))
42.0927851137247

Matrix Sine#

matA.Sinm(A)#

Calculates the sine function of the matrix.

See also: Wikipedia [1546], Wikipedia [1564].

See also: Eigen [216], Eigen [221].

The cosine of a square matrix \(A\) is defined in analogy with the matrix exponential.

\[\cos(A) = \frac{e^{iA} + e^{-iA}}{2}\]
\[\sin(A) = \frac{e^{iA} - e^{-iA}}{2i}\]
\[\cos^2(A) + \sin^2(A) = I\]

For real \(A\), we can write \(\cos(A) = \Re ( e^{iA} )\) and \(\sin(A) = \Im ( e^{iA} )\).

\[\cosh(A) = \frac{e^{A} + e^{-A}}{2}\]
\[\sinh(A) = \frac{e^{A} - e^{-A}}{2}\]

Examples:

>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> X = eye(3)
>>> cosm(X)
[0.54030230586814               0.0               0.0]
[             0.0  0.54030230586814               0.0]
[             0.0               0.0  0.54030230586814]
>>> X = hilbert(3)
>>> cosm(X)
[ 0.424403834569555  -0.316643413047167  -0.221474945949293]
[-0.316643413047167   0.820646708837824  -0.127183694770039]
[-0.221474945949293  -0.127183694770039   0.909236687217541]
>>> X = matrix([[1+j,-2],[0,-j]])
>>> cosm(X)
[(0.833730025131149 - 0.988897705762865j)  (1.07485840848393 - 0.17192140544213j)]
[                                     0.0               (1.54308063481524 + 0.0j)]

Matrix Cosine#

matA.Cosm()#

Calculates the cosine function of the matrix.

See also: Wikipedia [1546], Wikipedia [1564].

See also: Eigen [216], Eigen [214].

The cosine of a square matrix \(A\) is defined in analogy with the matrix exponential.

\[\cos(A) = \frac{e^{iA} + e^{-iA}}{2}\]
\[\sin(A) = \frac{e^{iA} - e^{-iA}}{2i}\]
\[\cos^2(A) + \sin^2(A) = I\]

For real \(A\), we can write \(\cos(A) = \Re ( e^{iA} )\) and \(\sin(A) = \Im ( e^{iA} )\).

\[\cosh(A) = \frac{e^{A} + e^{-A}}{2}\]
\[\sinh(A) = \frac{e^{A} - e^{-A}}{2}\]

Examples:

>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> X = eye(3)
>>> cosm(X)
[0.54030230586814               0.0               0.0]
[             0.0  0.54030230586814               0.0]
[             0.0               0.0  0.54030230586814]
>>> X = hilbert(3)
>>> cosm(X)
[ 0.424403834569555  -0.316643413047167  -0.221474945949293]
[-0.316643413047167   0.820646708837824  -0.127183694770039]
[-0.221474945949293  -0.127183694770039   0.909236687217541]
>>> X = matrix([[1+j,-2],[0,-j]])
>>> cosm(X)
[(0.833730025131149 - 0.988897705762865j)  (1.07485840848393 - 0.17192140544213j)]
[                                     0.0               (1.54308063481524 + 0.0j)]

Matrix Hyperbolic Sine#

matA.Sinhm()#

Calculates the hyperbolic sine function of the matrix.

See also: Wikipedia [1546], Wikipedia [1564].

See also: Eigen [216], Eigen [218].

The hyperbolic sine of a square matrix \(A\) is defined in analogy with the matrix exponential.

\[\sinh(A) = \frac{e^{A} - e^{-A}}{2}\]
\[\cosh(A) = \frac{e^{A} + e^{-A}}{2}\]
\[\cosh^2(A) + \sinh^2(A) = I\]
Sub DemoCplxMatrixFunctions()
    Console.WriteLine("Hello CplxMatrixFunctions!")
    Dim digits = 15
    Dim n As Int32 = 4
    Dim A, B, C, D, E, F As New cplx_mat_t
    A.RandomSymmetric(n)
    A.Print("A: ")

    B = A.SinhMat()
    B.Print("B = Sinh(A): ")
    C = A.CoshMat()
    C.Print("C = Cosh(A): ")
    D = C * C - B * B
    D.Print("C * C - B * B: ")
End Sub

Hello CplxMatrixFunctions!
A:
 0.254341-1.994995j,  1.110691+0.032289j, -0.311350-0.647176j,  0.547685-1.281594j;
 1.110691+0.032289j,  1.435774-1.303568j, -0.641133-0.284371j, -0.827662-0.328684j;
-0.311350-0.647176j, -0.641133-0.284371j, -0.217231+1.954100j, -0.776330-0.558306j;
 0.547685-1.281594j, -0.827662-0.328684j, -0.776330-0.558306j,  0.652181-1.335063j;


B = Sinh(A):
-0.344029-0.627502j,  0.956056-0.480516j, -0.134878-0.034162j, -0.590098+0.338244j;
 0.956056-0.480516j,  1.739419-3.149695j, -0.793048-0.141610j, -1.597165+0.360539j;
-0.134878-0.034162j, -0.793048-0.141610j, -0.152880+1.211517j, -0.351965+0.026714j;
-0.590098+0.338244j, -1.597165+0.360539j, -0.351965+0.026714j,  0.171917-0.584919j;

C = Cosh(A):
-0.039723+0.009230j,  0.532073-1.170311j, -0.476465+0.181642j, -1.605643-0.168838j;
 0.532073-1.170311j,  2.022858-2.326341j, -0.294411-0.149198j, -1.095355+1.019715j;
-0.476465+0.181642j, -0.294411-0.149198j, -0.216192+0.186437j,  0.187127+0.027892j;
-1.605643-0.168838j, -1.095355+1.019715j,  0.187127+0.027892j,  0.851060-0.052593j;

C * C - B * B:
 1.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j;
 0.000000+0.000000j,  1.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j;
 0.000000+0.000000j,  0.000000+0.000000j,  1.000000+0.000000j,  0.000000+0.000000j;
 0.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j,  1.000000+0.000000j;

Matrix Hyperbolic Cosine#

matA.Coshm()#

See also: Wikipedia [1546], Wikipedia [1564].

See also: Eigen [216], Eigen [217].

Calculates the hyperbolic cosine function of the matrix.

The hyperbolic cosine of a square matrix \(A\) is defined in analogy with the matrix exponential.

\[\cosh(A) = \frac{e^{A} + e^{-A}}{2}\]
\[\sinh(A) = \frac{e^{A} - e^{-A}}{2}\]
\[\cosh^2(A) + \sinh^2(A) = I\]
Sub DemoCplxMatrixFunctions()
    Console.WriteLine("Hello CplxMatrixFunctions!")
    Dim digits = 15
    Dim n As Int32 = 4
    Dim A, B, C, D, E, F As New cplx_mat_t
    A.RandomSymmetric(n)
    A.Print("A: ")
    B = A.ExpMat()
    B.Print("B = Exp(A): ")
    C = B.LogMat()
    C.Print("C = Log(B): ")

    D = B.SqrtMat()
    D.Print("D = Sqrt(B): ")
    E = D * D
    E.Print("E = D * D: ")

    B = A.SinMat()
    B.Print("B = Sin(A): ")
    C = A.CosMat()
    C.Print("C = Cos(A): ")
    D = B * B + C * C
    D.Print("B * B + C * C: ")

    B = A.SinhMat()
    B.Print("B = Sinh(A): ")
    C = A.CoshMat()
    C.Print("C = Cosh(A): ")
    D = C * C - B * B
    D.Print("C * C - B * B: ")
End Sub

Hello CplxMatrixFunctions!
A:
 0.254341-1.994995j,  1.110691+0.032289j, -0.311350-0.647176j,  0.547685-1.281594j;
 1.110691+0.032289j,  1.435774-1.303568j, -0.641133-0.284371j, -0.827662-0.328684j;
-0.311350-0.647176j, -0.641133-0.284371j, -0.217231+1.954100j, -0.776330-0.558306j;
 0.547685-1.281594j, -0.827662-0.328684j, -0.776330-0.558306j,  0.652181-1.335063j;

B = Exp(A):
-0.383752-0.618272j,  1.488129-1.650828j, -0.611343+0.147480j, -2.195741+0.169406j;
 1.488129-1.650828j,  3.762277-5.476036j, -1.087459-0.290808j, -2.692520+1.380254j;
-0.611343+0.147480j, -1.087459-0.290808j, -0.369072+1.397953j, -0.164837+0.054606j;
-2.195741+0.169406j, -2.692520+1.380254j, -0.164837+0.054606j,  1.022977-0.637512j;

C = Log(B):
 0.254341-1.994995j,  1.110691+0.032289j, -0.311350-0.647176j,  0.547685-1.281594j;
 1.110691+0.032289j,  1.435774-1.303568j, -0.641133-0.284371j, -0.827662-0.328684j;
-0.311350-0.647176j, -0.641133-0.284371j, -0.217231+1.954100j, -0.776330-0.558306j;
 0.547685-1.281594j, -0.827662-0.328684j, -0.776330-0.558306j,  0.652181-1.335063j;

D = Sqrt(B):
 0.508620-0.931943j,  0.661969-0.319593j, -0.337003-0.172068j, -0.485698-0.655636j;
 0.661969-0.319593j,  2.077099-1.250104j, -0.401242-0.208424j, -0.746596+0.018077j;
-0.337003-0.172068j, -0.401242-0.208424j,  0.502520+0.963369j, -0.377613-0.193978j;
-0.485698-0.655636j, -0.746596+0.018077j, -0.377613-0.193978j,  1.013700-0.687531j;

E = D * D:
-0.383752-0.618272j,  1.488129-1.650828j, -0.611343+0.147480j, -2.195741+0.169406j;
 1.488129-1.650828j,  3.762277-5.476036j, -1.087459-0.290808j, -2.692520+1.380254j;
-0.611343+0.147480j, -1.087459-0.290808j, -0.369072+1.397953j, -0.164837+0.054606j;
-2.195741+0.169406j, -2.692520+1.380254j, -0.164837+0.054606j,  1.022977-0.637512j;

B = Sin(A):
 3.185996-4.423204j,  1.572018+0.576777j, -0.236161-2.007119j,  3.744383-3.392271j;
 1.572018+0.576777j,  0.829215-0.073722j, -0.452703-0.362702j,  0.213357-0.343298j;
-0.236161-2.007119j, -0.452703-0.362702j, -0.274184+2.749772j, -0.899405-1.828027j;
 3.744383-3.392271j,  0.213357-0.343298j, -0.899405-1.828027j,  3.157659-2.589512j;

C = Cos(A):
 4.524926+3.229556j, -0.735268+1.275592j,  1.749846+0.069384j,  3.245352+3.579443j;
-0.735268+1.275592j, -0.328697+0.628146j,  0.172580+0.683175j,  0.290239+0.309901j;
 1.749846+0.069384j,  0.172580+0.683175j,  3.469410-0.664084j,  0.996030+0.538410j;
 3.245352+3.579443j,  0.290239+0.309901j,  0.996030+0.538410j,  2.603737+2.713445j;

B * B + C * C:
 1.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j;
 0.000000+0.000000j,  1.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j;
 0.000000+0.000000j,  0.000000+0.000000j,  1.000000+0.000000j,  0.000000+0.000000j;
 0.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j,  1.000000+0.000000j;

B = Sinh(A):
-0.344029-0.627502j,  0.956056-0.480516j, -0.134878-0.034162j, -0.590098+0.338244j;
 0.956056-0.480516j,  1.739419-3.149695j, -0.793048-0.141610j, -1.597165+0.360539j;
-0.134878-0.034162j, -0.793048-0.141610j, -0.152880+1.211517j, -0.351965+0.026714j;
-0.590098+0.338244j, -1.597165+0.360539j, -0.351965+0.026714j,  0.171917-0.584919j;

C = Cosh(A):
-0.039723+0.009230j,  0.532073-1.170311j, -0.476465+0.181642j, -1.605643-0.168838j;
 0.532073-1.170311j,  2.022858-2.326341j, -0.294411-0.149198j, -1.095355+1.019715j;
-0.476465+0.181642j, -0.294411-0.149198j, -0.216192+0.186437j,  0.187127+0.027892j;
-1.605643-0.168838j, -1.095355+1.019715j,  0.187127+0.027892j,  0.851060-0.052593j;

C * C - B * B:
 1.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j;
 0.000000+0.000000j,  1.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j;
 0.000000+0.000000j,  0.000000+0.000000j,  1.000000+0.000000j,  0.000000+0.000000j;
 0.000000+0.000000j,  0.000000+0.000000j,  0.000000+0.000000j,  1.000000+0.000000j;

Matrix Square Root#

matA.Sqrtm()#

Computes a square root of the square matrix \(A\), i.e. returns a matrix \(B = A^{1/2}\) such that \(B^2 = A\). The square root of a matrix, if it exists, is not unique.

See also: Wikipedia [1546], Wikipedia [1563].

See also: Eigen [216], Eigen [222].

Examples

Square roots of some simple matrices:

>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> sqrtm([[1,0], [0,1]])
[1.0  0.0]
[0.0  1.0]
>>> sqrtm([[0,0], [0,0]])
[0.0  0.0]
[0.0  0.0]
>>> sqrtm([[2,0],[0,1]])
[1.4142135623731  0.0]
[            0.0  1.0]
>>> sqrtm([[1,1],[1,0]])
[ (0.920442065259926 - 0.21728689675164j)  (0.568864481005783 + 0.351577584254143j)]
[(0.568864481005783 + 0.351577584254143j)  (0.351577584254143 - 0.568864481005783j)]
>>> sqrtm([[1,0],[0,1]])
[1.0  0.0]
[0.0  1.0]
>>> sqrtm([[-1,0],[0,1]])
[(0.0 - 1.0j)           0.0]
[         0.0  (1.0 + 0.0j)]
>>> sqrtm([[j,0],[0,j]])
[(0.707106781186547 + 0.707106781186547j)                                       0.0]
[                                     0.0  (0.707106781186547 + 0.707106781186547j)]

A square root of a rotation matrix, giving the corresponding half-angle rotation matrix:

>>> t1 = 0.75
>>> t2 = t1 * 0.5
>>> A1 = matrix([[cos(t1), -sin(t1)], [sin(t1), cos(t1)]])
>>> A2 = matrix([[cos(t2), -sin(t2)], [sin(t2), cos(t2)]])
>>> sqrtm(A1)
[0.930507621912314  -0.366272529086048]
[0.366272529086048   0.930507621912314]
>>> A2
[0.930507621912314  -0.366272529086048]
[0.366272529086048   0.930507621912314]

The identity \((A^2)^{1/2} = A\) does not necessarily hold:

>>> A = matrix([[4,1,4],[7,8,9],[10,2,11]])
>>> sqrtm(A**2)
[ 4.0  1.0   4.0]
[ 7.0  8.0   9.0]
[10.0  2.0  11.0]
>>> sqrtm(A)**2
[ 4.0  1.0   4.0]
[ 7.0  8.0   9.0]
[10.0  2.0  11.0]
>>> A = matrix([[-4,1,4],[7,-8,9],[10,2,11]])
>>> sqrtm(A**2)
[  7.43715112194995  -0.324127569985474   1.8481718827526]
[-0.251549715716942    9.32699765900402  2.48221180985147]
[  4.11609388833616   0.775751877098258   13.017955697342]
>>> chop(sqrtm(A)**2)
[-4.0   1.0   4.0]
[ 7.0  -8.0   9.0]
[10.0   2.0  11.0]

For some matrices, a square root does not exist:

>>> sqrtm([[0,1], [0,0]])
Traceback (most recent call last):
    ...
ZeroDivisionError: matrix is numerically singular

Two examples from the documentation for Matlab’s sqrtm:

>>> mp.dps = 15; mp.pretty = True
>>> sqrtm([[7,10],[15,22]])
[1.56669890360128  1.74077655955698]
[2.61116483933547  4.17786374293675]
>>>
>>> X = matrix(\
...   [[5,-4,1,0,0],
...   [-4,6,-4,1,0],
...   [1,-4,6,-4,1],
...   [0,1,-4,6,-4],
...   [0,0,1,-4,5]])
>>> Y = matrix(\
...   [[2,-1,-0,-0,-0],
...   [-1,2,-1,0,-0],
...   [0,-1,2,-1,0],
...   [-0,0,-1,2,-1],
...   [-0,-0,-0,-1,2]])
>>> mnorm(sqrtm(X) - Y)
4.53155328326114e-19

Matrix Logarithm#

matA.Logm()#

Calculates the logarithm of the matrix.

See also: Wikipedia [1546], Wikipedia [1554].

See also: Eigen [216], Eigen [219].

Computes a logarithm of the square matrix \(A\), i.e. returns a matrix \(B = \log(A)\) such that \(\exp(B) = A\). The logarithm of a matrix, if it exists, is not unique.

Examples

Logarithms of some simple matrices:

>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> X = eye(3)
>>> logm(X)
[0.0  0.0  0.0]
[0.0  0.0  0.0]
[0.0  0.0  0.0]
>>> logm(2*X)
[0.693147180559945                0.0                0.0]
[              0.0  0.693147180559945                0.0]
[              0.0                0.0  0.693147180559945]
>>> logm(expm(X))
[1.0  0.0  0.0]
[0.0  1.0  0.0]
[0.0  0.0  1.0]

A logarithm of a complex matrix:

>>> X = matrix([[2+j, 1, 3], [1-j, 1-2*j, 1], [-4, -5, j]])
>>> B = logm(X)
>>> nprint(B)
[ (0.808757 + 0.107759j)    (2.20752 + 0.202762j)   (1.07376 - 0.773874j)]
[ (0.905709 - 0.107795j)  (0.0287395 - 0.824993j)  (0.111619 + 0.514272j)]
[(-0.930151 + 0.399512j)   (-2.06266 - 0.674397j)  (0.791552 + 0.519839j)]
>>> chop(expm(B))
[(2.0 + 1.0j)           1.0           3.0]
[(1.0 - 1.0j)  (1.0 - 2.0j)           1.0]
[        -4.0          -5.0  (0.0 + 1.0j)]

A matrix \(X\) close to the identity matrix, for which \(\log(\exp(X)) = \exp(\log(X)) = X\) holds:

>>> X = eye(3) + hilbert(3)/4
>>> X
[              1.25             0.125  0.0833333333333333]
[             0.125  1.08333333333333              0.0625]
[0.0833333333333333            0.0625                1.05]
>>> logm(expm(X))
[              1.25             0.125  0.0833333333333333]
[             0.125  1.08333333333333              0.0625]
[0.0833333333333333            0.0625                1.05]
>>> expm(logm(X))
[              1.25             0.125  0.0833333333333333]
[             0.125  1.08333333333333              0.0625]
[0.0833333333333333            0.0625                1.05]

A logarithm of a rotation matrix, giving back the angle of the rotation:

>>> t = 3.7
>>> A = matrix([[cos(t),sin(t)],[-sin(t),cos(t)]])
>>> chop(logm(A))
[             0.0  -2.58318530717959]
[2.58318530717959                0.0]
>>> (2*pi-t)
2.58318530717959

For some matrices, a logarithm does not exist:

>>> logm([[1,0], [0,0]])
Traceback (most recent call last):
    ...
ZeroDivisionError: matrix is numerically singular

Logarithm of a matrix with large entries:

>>> logm(hilbert(3) * 10**20).apply(re)
[ 45.5597513593433  1.27721006042799  0.317662687717978]
[ 1.27721006042799  42.5222778973542   2.24003708791604]
[0.317662687717978  2.24003708791604    42.395212822267]

Matrix power#

mat.Powm(r)#

Computes \(A^r = \exp(A \log r)\) for a matrix \(A\) and complex number \(r\).

See also: Wikipedia [1546], Wikipedia [1556], Wikipedia [1554].

See also: Eigen [216], Eigen [220].

Examples

Powers and inverse powers of a matrix:

>>> from mpmath import *
>>> mp.dps = 15; mp.pretty = True
>>> A = matrix([[4,1,4],[7,8,9],[10,2,11]])
>>> powm(A, 2)
[ 63.0  20.0   69.0]
[174.0  89.0  199.0]
[164.0  48.0  179.0]
>>> chop(powm(powm(A, 4), 1/4.))
[ 4.0  1.0   4.0]
[ 7.0  8.0   9.0]
[10.0  2.0  11.0]
>>> powm(extraprec(20)(powm)(A, -4), -1/4.)
[ 4.0  1.0   4.0]
[ 7.0  8.0   9.0]
[10.0  2.0  11.0]
>>> chop(powm(powm(A, 1+0.5j), 1/(1+0.5j)))
[ 4.0  1.0   4.0]
[ 7.0  8.0   9.0]
[10.0  2.0  11.0]
>>> powm(extraprec(5)(powm)(A, -1.5), -1/(1.5))
[ 4.0  1.0   4.0]
[ 7.0  8.0   9.0]
[10.0  2.0  11.0]

A Fibonacci-generating matrix:

>>> powm([[1,1],[1,0]], 10)
[89.0  55.0]
[55.0  34.0]
>>> fib(10)
55.0
>>> powm([[1,1],[1,0]], 6.5)
[(16.5166626964253 - 0.0121089837381789j)  (10.2078589271083 + 0.0195927472575932j)]
[(10.2078589271083 + 0.0195927472575932j)  (6.30880376931698 - 0.0317017309957721j)]
>>> (phi**6.5 - (1-phi)**6.5)/sqrt(5)
(10.2078589271083 - 0.0195927472575932j)
>>> powm([[1,1],[1,0]], 6.2)
[ (14.3076953002666 - 0.008222855781077j)  (8.81733464837593 + 0.0133048601383712j)]
[(8.81733464837593 + 0.0133048601383712j)  (5.49036065189071 - 0.0215277159194482j)]
>>> (phi**6.2 - (1-phi)**6.2)/sqrt(5)
(8.81733464837593 - 0.0133048601383712j)
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