Python cheatsheet

Operators

Command Description
* multiplication operation: 2*3 returns 6
** power operation: 2**3 returns 8
@

matrix multiplication:

import numpy as np
A = np.array([[1,2,3]])
B = np.array([[3],[2],[1]])
A @ B

returns

array([[10]])

Data Types

Command Description
l = [a1,a2,...,an] Constructs a list containing the objects \(a1,a2,...,an\). You can append to the list using l.append(). The \(ith\) element of \(l\) can be accessed using l[i]
t =(a1,a2,...,an) Constructs a tuple containing the objects \(a1,a2,...,an\). The \(ith\) element of \(t\) can be accessed using t[i]

Built-In Functions

Command Description
len(iterable)

len is a function that takes an iterable, such as a list, tuple or numpy array and returns the number of items in that object. For a numpy array, len returns the length of the outermost dimension

len(np.zeros((5,4)))

returns 5.

zip

Make an iterator that aggregates elements from each of the iterables.

x = [1, 2, 3]
y = [4, 5, 6]
zipped = zip(x, y)
list(zipped)

returns [(1, 4), (2, 5), (3, 6)]

Iterating

Command Description
for a in iterable:

For loop used to perform a sequence of commands (denoted using tabs) for each element in an iterable object such as a list, tuple, or numpy array. An example code is

l  = []
for i in [1,2,3]:
    l.append(i**2)
print(l)

prints [1,4,9]

Comparisons and Logical Operators

Command Description
if condition:

Performs code if a condition is met (using tabs). For example

if x == 5:
    x = x**2
else:
    x = x**3

squares \(x\) if \(x\) is \(5\), otherwise cubes it.

User-Defined Functions

Command Description
lambda

Used for create anonymous one line functions of the form:

f = lambda x,y: 5*x+y

The code after the lambda but before variables specifies the parameters. The code after the colon tells python what object to return.

def

The def command is used to create functions of more than one line:

def g(x,y):
    """
    Docstring
    """
    ret = sin(x)
    return ret + y

The code immediately following def names the function, in this example g . The variables in the parenthesis are the parameters of the function. The remaining lines of the function are denoted by tab indents. The return statement specifies the object to be returned.

Numpy

Command Description
np.array(object,dtype = None)

np.array constructs a numpy array from an object, such as a list or a list of lists. dtype allows you to specify the type of object the array is holding. You will generally note need to specify the dtype. Examples:

np.array([1, 2, 3]) #creates 1 dim array of ints
np.array( [1,2,3.0] )#creates 1 dim array of floats
np.array( [ [1,2],[3,4] ]) #creates a 2 dim array
A[i1,i2,...,in]

Access a the element in numpy array A in with index i1 in dimension 1, i2 in dimension 2, etc. Can use : to access a range of indices, where imin:imax represents all \(i\) such that \(imin \leq i < imax\). Always returns an object of minimal dimension. For example,

A[:,2]

returns the 2nd column (counting from 0) of A as a 1 dimensional array and

A[0:2,:]

returns the 0th and 1st rows in a 2 dimensional array.

np.zeros(shape)

Constructs numpy array of shape shape. Here shape is an integer of sequence of integers. Such as 3, (1,2),(2,1), or (5,5). Thus

np.zeros((5,5))

Constructs an \(5\times 5\) array while

np.zeros(5,5)

will throw an error.

np.ones(shape) Same as np.zeros but produces an array of ones
np.linspace(a,b,n)

Returns a numpy array with \(n\) linearly spaced points between \(a\) and \(b\). For example

np.linspace(1,2,10)

returns

array([ 1.        ,  1.11111111,  1.22222222,  1.33333333,
1.44444444, 1.55555556,  1.66666667,  1.77777778,
1.88888889,  2.        ])
np.eye(N)

Constructs the identity matrix of size \(N\). For example

np.eye(3)

returns the \(3\times 3\) identity matrix:

\[\begin{split}\left(\begin{matrix}1&0&0\\0&1&0\\ 0&0&1\end{matrix}\right)\end{split}\]
np.diag(a)

np.diag has 2 uses. First if a is a 2 dimensional array then np.diag returns the principle diagonal of the matrix. Thus

np.diag( [ [1,3], [5,6] ])

returns [1,6].

If \(a\) is a 1 dimensional array then np.diag constructs an array with $a$ as the principle diagonal. Thus,

np.diag([1,2])

returns

\[\begin{split}\left(\begin{matrix}1&0\\0&2\end{matrix}\right)\end{split}\]
np.random.rand(d0, d1, ..., dn)

Constructs a numpy array of shape (d0,d1,...,dn) filled with random numbers drawn from a uniform distribution between :math`(0,1)`. For example, np.random.rand(2,3) returns

array([[ 0.69060674,  0.38943021,  0.19128955],
[ 0.5419038 ,  0.66963507,  0.78687237]])
np.random.randn(d0, d1, ..., dn) Same as np.random.rand(d0, d1, ..., dn) except that it draws from the standard normal distribution \(\mathcal N(0,1)\) rather than the uniform distribution.
A.T Reverses the dimensions of an array (transpose). For example, if \(x = \left(\begin{matrix} 1& 2\\3&4\end{matrix}\right)\) then x.T returns \(\left(\begin{matrix} 1& 3\\2&4\end{matrix}\right)\)
np.hstack(tuple)

Take a sequence of arrays and stack them horizontally to make a single array. For example

a = np.array(( [1,2,3] )
b = np.array( [2,3,4] )
np.hstack( (a,b) )

returns [1,2,3,2,3,4] while

a = np.array( [[1],[2],[3]] )
b = np.array( [[2],[3],[4]] )
np.hstack((a,b))

returns \(\left( \begin{matrix} 1&2\\2&3\\ 3&4 \end{matrix}\right)\)

np.vstack(tuple)

Like np.hstack. Takes a sequence of arrays and stack them vertically to make a single array. For example

a = np.array( [1,2,3] )
b = np.array( [2,3,4] )
np.hstack( (a,b) )

returns

array( [ [1,2,3],
[2,3,4] ] )
np.amax(a, axis = None)

By default np.amax(a) finds the maximum of all elements in the array \(a\). Can specify maximization along a particular dimension with axis. If

a = np.array( [ [2,1], [3,4] ]) #creates a 2 dim array

then

np.amax(a,axis = 0) #maximization along row (dim 0)

returns array([3,4]) and

np.amax(a, axis = 1) #maximization along column (dim 1)

returns array([2,4])

np.amin(a, axis = None) Same as np.amax except returns minimum element.
np.argmax(a, axis = None)

Performs similar function to np.amax except returns index of maximal element. By default gives index of flattened array, otherwise can use axis to specify dimension. From the example for np.amax

np.amax(a,axis = 0) #maximization along row (dim 0)

returns array([1,1]) and

np.amax(a, axis = 1) #maximization along column (dim 1)

returns array([0,1])

np.argmin(a, axis =None) Same as np.argmax except finds minimal index.
np.dot(a,b) or a.dot(b) Returns an array equal to the dot product of \(a\) and \(b\). For this operation to work the innermost dimension of \(a\) must be equal to the outermost dimension of \(b\). If \(a\) is a \((3,2)\) array and \(b\) is a \((2)\) array then np.dot(a,b) is valid. If \(b\) is a \((1,2)\) array then the operation will return an error.

numpy.linalg

Command Description
np.linalg.inv(A)

For a 2-dimensional array \(A\). np.linalg.inv returns the inverse of \(A\). For example, for a \((2,2)\) array \(A\)

np.linalg.inv(A).dot(A)

returns

np.array( [1,0],
[0,1] ])
np.linalg.eig(A)

Returns a 1-dimensional array with all the eigenvalues of $A$ as well as a 2-dimensional array with the eigenvectors as columns. For example,

eigvals,eigvecs = np.linalg.eig(A)

returns the eigenvalues in eigvals and the eigenvectors in eigvecs. eigvecs[:,i] is the eigenvector of \(A\) with eigenvalue of eigval[i].

np.linalg.solve(A,b)

Constructs array \(x\) such that A.dot(x) is equal to \(b\). Theoretically should give the same answer as

Ainv = np.linalg.inv(A)
x = Ainv.dot(b)

but numerically more stable.

Pandas

Command Description
pd.Series()

Constructs a Pandas Series Object from some specified data and/or index

s1 = pd.Series([1,2,3])
s2 = pd.Series([1,2,3], index=['a','b','c'])
pd.DataFrame()

Constructs a Pandas DataFrame object from some specified data and/or index, column names etc.

d = {'a' : [1,2,3], 'b' : [4,5,6]}
df = pd.DataFrame(d)

or alternatively,

a = [1,2,3]
b = [4,5,6]
df = pd.DataFrame(list(zip(a,b)), columns=['a','b'])

Plotting

Command Description
plt.plot(x,y,s =None)

The plot command is included in matplotlib.pyplot. The plot command is used to plot \(x\) versus \(y\) where \(x\) and \(y\) are iterables of the same length. By default the plot command draws a line, using the \(s\) argument you can specify type of line and color. For example ‘-‘,’- -‘,’:’,’o’,’x’, and ‘-o’ reprent line, dashed line, dotted line, circles, x’s, and circle with line through it respectively. Color can be changed by appending ‘b’,’k’,’g’ or ‘r’, to get a blue, black, green or red plot respectively. For example,

import numpy as np
import matplotlib.pyplot as plt
x=np.linspace(0,10,100)
N=len(x)
v= np.cos(x)
plt.figure(1)
plt.plot(x,v,'-og')
plt.show()
plt.savefig('tom_test.eps')

plots the cosine function on the domain (0,10) with a green line with circles at the points \(x,v\)