Inserting complex functions in a python code

Question

I have been trying to insert $e^ix$ as matrix element. The main aim is to find the eigenvalue of a matrix which has many complex functions as elements. Can anyone help me how to insert it? My failed attempt is below:

for i in range(0,size):
                    
            H[i,i]=-2*(cmath.exp((i+1)*aj))
            H[i,i+1]=1.0
            H[i,i-1]=1.0

'a' is defined earlier in the program. The error flagged shows that aj is not defined. Using cmath I thought a complex number can be expontiated as (x+yj). Unfortunately, I couldn't figure out the right way to use it. Any help would be appreciated

What type is ```a```? If you're trying to use the imaginary component of ```a``` you can refer to it as ```a.imag```. — sj95126, Aug 08 '21 at 03:44
What is `H`, `a `? In python `1j` is imaginary. `2+4j` is complex. A numpy array has to be created with complex elements, or initialized wth complex dtype. — hpaulj, Aug 08 '21 at 03:58
https://numpy.org/doc/stable/reference/generated/numpy.exp.html has a complex example — hpaulj, Aug 08 '21 at 04:05
H is a hamiltonian matrix which is a square matrix of chosen dimension. a values will be picked from an array I define separately. The array consists only real numbers a. Something like $H_i,i=-i*exp(aj)$ where a is to be picked from the array and exponentiated to be an Euler number. @hpaulj — anand_quanta, Aug 08 '21 at 04:44
I'm more interested in a numpy style description - shape, dtype, what commands you used to create it. 'hamiltonian' does not match anything in the numpy docs. — hpaulj, Aug 08 '21 at 07:01

hpaulj · Accepted Answer · 2021-08-08T15:19:18.997

Define a small float array:

In [214]: H = np.eye(3)
In [215]: H
Out[215]: 
array([[1., 0., 0.],
       [0., 1., 0.],
       [0., 0., 1.]])

Create a complex number:

In [216]: 1+3j
Out[216]: (1+3j)
In [217]: np.exp(1+3j)
Out[217]: (-2.6910786138197937+0.383603953541131j)

Trying to assign it to H:

In [218]: H[1,1]=np.exp(1+3j)
<ipython-input-218-6c0b228d2833>:1: ComplexWarning: Casting complex values to real discards the imaginary part
  H[1,1]=np.exp(1+3j)
In [219]: H
Out[219]: 
array([[ 1.        ,  0.        ,  0.        ],
       [ 0.        , -2.69107861,  0.        ],
       [ 0.        ,  0.        ,  1.        ]])

Now make an complex dtype array:

In [221]: H = np.eye(3).astype( complex)
In [222]: H[1,1]=np.exp(1+3j)
In [223]: H
Out[223]: 
array([[ 1.        +0.j        ,  0.        +0.j        ,
         0.        +0.j        ],
       [ 0.        +0.j        , -2.69107861+0.38360395j,
         0.        +0.j        ],
       [ 0.        +0.j        ,  0.        +0.j        ,
         1.        +0.j        ]])

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For an array of values:

In [225]: a = np.array([1,2,3])
In [226]: np.exp(a+1j*a)
Out[226]: 
array([  1.46869394+2.28735529j,  -3.07493232+6.7188497j ,
       -19.88453084+2.83447113j])

In [228]: H[:,0]=np.exp(a+1j*a)
In [229]: H
Out[229]: 
array([[  1.46869394+2.28735529j,   0.        +0.j        ,
          0.        +0.j        ],
       [ -3.07493232+6.7188497j ,  -2.69107861+0.38360395j,
          0.        +0.j        ],
       [-19.88453084+2.83447113j,   0.        +0.j        ,
          1.        +0.j        ]])

Thanks a lot @hpaulj. This clears up one part of my problem. However, I am wondering if you could kindly help me to understand how to do the next part. I have a 1D matrix 'a' which has x number of elements. I want to insert each of these x elements in matrix H which is a x*x matrix. — anand_quanta, Aug 08 '21 at 14:56

Inserting complex functions in a python code

1 Answers1

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