numpy cheat sheet

General

Matrix construction

TODO (Hannah): for dense matrices (matrix vs. array) as well as sparse matrices (csr_matrix((data, indices, indptr))

Accessing elements

TODO (Hannah): crazy element access magic, single elements, entire rows, sub-matrices

Matrix operations

TODO (Raghu): examples of dot product (dense * dense, dense * sparse, sparse * sparse), usage of both matrix.dot() and * (and how it behaves in different contexts), constant factor adding / multiplication

TODO (Claudius): Element-wise operations like taking log, sqrt. Multiplying two m*n matrices element-wise (for example, to square the entries in a matrix etc...)

Row- or column-wise operations

TODO (Claudius): summing of rows or columns, sorting rows / columns etc

Useful methods

TODO (Natalie): numpy.where, numpy.argsort, numpy.min, numpy.argmin, numpy.round (useful for tests)

Special matrices

Diagonal matrix

Matrix (usually square) in which all entries are zero, except on the main diagonal. Use numpy.diag to either create a diagonal matrix from a givin main diagonal, or extract the diagonal matrix from a given matrix.

>>> numpy.diag([1,2,3])
array([[1, 0, 0],
       [0, 2, 0],
       [0, 0, 3]])

>>> numpy.diag([[1, 5, 4],
                [7, 2, 4],
                [4, 7, 3]])
array([1, 2, 3])

For a sparse matrix, use scipy.spare.spdiags

Identity matrix

Special diagonal m*m matrix where all elements on the main diagonal are 1. Sometimes denoted as 1. Read as the '1' of matrix world. For example, a n*m matrix A multiplied with an m*m identity matrix yields A again. Use numpy.identity(k) to create a k*k identity matrix.

>>> numpy.identity(4)
array([[ 1.,  0.,  0.,  0.],
       [ 0.,  1.,  0.,  0.],
       [ 0.,  0.,  1.,  0.],
       [ 0.,  0.,  0.,  1.]])

>>> numpy.array([[1, 2, 3],
                 [3, 4, 3]]).dot(numpy.identity(3))
array([[ 1.,  2.,  3.],
       [ 3.,  4.,  3.]])

For a sparse matrix, use scipy.sparse.identity

Triangular matrix

A (square) matrix where all elements below (upper triangle) or above (lower triangle) the main diagonal are zero. numpy.triu creates the upper (u), numpy.triu the lower (l) triangular matrix from a given matrix.

>>> numpy.triu([[1, 5, 4],
                [7, 2, 4],
                [4, 7, 3]])
array([[1, 5, 4],
       [0, 2, 4],
       [0, 0, 3]])

>>> numpy.tril([[1, 5, 4],
                [7, 2, 4],
                [4, 7, 3]])
array([[1, 0, 0],
       [7, 2, 0],
       [4, 7, 3]])

For a sparse matrix, use scipy.sparse.triu and scipy.sparse.tril

Matrix decomposition

Singular Value Decompostion (SVD)

Factorize a matrix A (m*n) into three matrices U (m * r), S (r * r) and V (r * n) such that A = U * S * V. Here r is the rank of A.

Use numpy.linalg.svd to do a singular value decomposition for a dense matrix. Use scipy.sparse.linalg.svds for sparse matrices (computes the largest k singular values for a sparse matrix).

>>> Uk, Sk, Vk = svds(csr_matrix([[1, 2, 3], [3, 4, 5], [5, 6, 4]], dtype=float), 2)
>>> print("Uk:\n", Uk, "\nSk:\n", Sk, "\nVk:\n", Vk)
U:
 [[ 0.56475636 -0.30288472]
 [ 0.51457155 -0.59799935]
 [-0.64518709 -0.74206309]] 
S:
 [  2.13530566  11.67829513] 
V:
 [[-0.52332762 -0.32001209  0.78975975]
 [-0.49726421 -0.63794803 -0.58800563]]