#acl All:read,write

= NumPy Cheat Sheet =

== General ==

TODO (?)

== 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 ==

=== Constant addition ===
Addition of a constant adds it to every element of the matrix (only for dense matrices)

{{{
>>> B_dense = numpy.matrix([[2, 1], [3, 4]], dtype=float)
>>> B_dense + 10
matrix([[ 12.,  11.],
        [ 13.,  14.]])
}}}

=== Multiplication by a constant ===
Multiplication by a constant multiplies every element of the matrix by that constant (both for sparse and dense matrices)

{{{
>>> A_sparse = csr_matrix([[1, 0], [0, 1], [3, 2]], dtype=float)
>>> (A_sparse * 10).todense()
matrix([[ 10.,   0.],
        [  0.,  10.],
        [ 30.,  20.]])
}}}

=== Multiplication ===
* produces the '''normal''' matrix multiplication between a csr_matrix (sparse) and a numpy matrix (dense).<<BR>>
* produces the '''element-wise''' matrix multiplication for numpy arrays (also dense). In these cases Python broadcasts the operands in case their dimensions mismatch.

matrix.dot() produces the normal matrix multiplication between a csr_matrix and a numpy matrix '''except''' in the case of a dense.dot(sparse) matrix multiplication.

The result of a matrix multiplication between:
 *a sparse and a sparse matrix is sparse
 *a sparse and a dense matrix is dense
 *a dense and a dense matrix is dense

https://docs.scipy.org/doc/scipy/reference/sparse.html <<BR>>
http://www.scipy-lectures.org/intro/numpy/operations.html

{{{
>>> A_sparse = csr_matrix([[1, 0], [0, 1], [3, 2]], dtype=float)
>>> B_dense = numpy.matrix([[2, 1], [3, 4]], dtype=float)

>>> A_dense = A_sparse.todense()
>>> B_sparse = csr_matrix(B_dense)


## Sparse with sparse
>>> C_sparse = A_sparse * B_sparse #(Normal Matrix multiplication, 3x2 matrix with 2x2 matrix)
>>> C_sparse.todense()
matrix([[  2.,   1.],
        [  3.,   4.],
        [ 12.,  11.]])

>>> C_sparse = A_sparse.dot(B_sparse) #(Normal Matrix multiplication, 3x2 matrix with 2x2 matrix)
>>> C_sparse.todense()
matrix([[  2.,   1.],
        [  3.,   4.],
        [ 12.,  11.]])


## Sparse with dense
>>> C_dense = A_sparse * B_dense #(Normal Matrix multiplication, 3x2 matrix with 2x2 matrix)
>>> C_dense
matrix([[  2.,   1.],
        [  3.,   4.],
        [ 12.,  11.]])
>>> C_dense = A_sparse.dot(B_dense) #(Normal Matrix multiplication, 3x2 matrix with 2x2 matrix)
>>> C_dense
matrix([[  2.,   1.],
        [  3.,   4.],
        [ 12.,  11.]])


## Dense with sparse
>>> C_dense = A_dense * B_sparse
>>> C_dense
matrix([[  2.,   1.],
        [  3.,   4.],
        [ 12.,  11.]])

>>> A_dense.dot(B_sparse)
matrix([[ <2x2 sparse matrix of type '<class 'numpy.float64'>'
 	with 4 stored elements in Compressed Sparse Row format>,
         <2x2 sparse matrix of type '<class 'numpy.float64'>'
 	with 4 stored elements in Compressed Sparse Row format>],
        [ <2x2 sparse matrix of type '<class 'numpy.float64'>'
 	with 4 stored elements in Compressed Sparse Row format>,
         <2x2 sparse matrix of type '<class 'numpy.float64'>'
 	with 4 stored elements in Compressed Sparse Row format>],
        [ <2x2 sparse matrix of type '<class 'numpy.float64'>'
 	with 4 stored elements in Compressed Sparse Row format>,
         <2x2 sparse matrix of type '<class 'numpy.float64'>'
 	with 4 stored elements in Compressed Sparse Row format>]], dtype=object)


## Dense with dense
>>> C_dense = A_dense.dot(B_dense) #(Normal Matrix multiplication, 3x2 matrix with 2x2 matrix)
>>> C_dense
matrix([[  2.,   1.],
        [  3.,   4.],
        [ 12.,  11.]])
>>> C_dense = A_dense * B_dense #(Normal Matrix multiplication, 3x2 matrix with 2x2 matrix)
>>> C_dense
matrix([[  2.,   1.],
        [  3.,   4.],
        [ 12.,  11.]])

}}}

{{{
## numpy.ndarray
>>> A_ndarray = numpy.array([[1, 0], [0, 1], [3, 2]])
>>> B_ndarray = numpy.array([[2, 1], [3, 4]])
>>> C_ndarray = numpy.array([2, 1])

>>> B_ndarray * B_ndarray #(Element-wise Matrix multiplication, 2x2 matrix with 2x2 matrix)
array([[ 4,  1],
       [ 9, 16]])
>>> B_ndarray.dot(B_ndarray) #(Normal Matrix multiplication, 2x2 matrix with 2x2 matrix)
array([[ 7,  6],
       [18, 19]])
>>> A_ndarray.dot(B_ndarray) #(Normal Matrix multiplication, 3x2 matrix with 2x2 matrix)
array([[ 2,  1],
       [ 3,  4],
       [12, 11]])
>>> C_ndarray * B_ndarray #(Broadcasting)
array([[4, 1],
       [6, 4]])
}}}

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 ==

=== numpy.round ===
Takes an array and rounds its values to the given number of decimals. Note that for values exactly halfway between rounded decimal values, Numpy rounds to the nearest even value. [[https://docs.scipy.org/doc/numpy-1.10.0/reference/generated/numpy.around.html|numpy.around]]

{{{
>>> numpy.round([1.98, 2.34, 4.76], 1)
[ 2.   2.3  4.8]
}}}
{{{
>>> numpy.round([1.5, 0.5, 3.5, 4.5], 0)
[ 2.  0.  4.  4.]
}}}

=== numpy.min ===
Takes an array and returns its minimum value. If an axis is specified, returns the minimum along the axis. [[https://docs.scipy.org/doc/numpy/reference/generated/numpy.amin.html|numpy.amin]]
{{{
>>> numpy.min([[5, 0, 1], [4, 3, 2]])
0
}}}
{{{
>>> numpy.min([[5, 0, 1], [4, 3, 2]], axis=0)
[4 0 1]
}}}

=== numpy.argmin ===
Takes an array and returns the index of the minimum value of the flattened array. If an axis is specified, returns the indices of the minimum values along the axis. [[https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmin.html|numpy.argmin]]
{{{
>>> numpy.argmin([[5, 0, 1], [4, 3, 2]])
1
}}}
{{{
>>> numpy.argmin([[5, 0, 1], [4, 3, 2]], axis=0)
[1 0 0]
}}}

=== numpy.argsort ===
Takes an array a and returns an array of indices that sort a. Optionally, you can specify the axis along which a will be sorted. By default the axis is -1. [[https://docs.scipy.org/doc/numpy/reference/generated/numpy.argsort.html|numpy.argsort]]

{{{
>>> numpy.argsort([[0, 4, 0], [4, 3, 2]], axis=0)
[[0 1 0]
 [1 0 1]]
}}}
{{{
>>> numpy.argsort([[0, 4, 0], [4, 3, 2]], axis=1)
[[0 2 1]
 [2 1 0]]
}}}

=== numpy.where ===
Takes a condition and optionally two array-like objects x and y. If x and y are specified, returns an array that contains elements from x where condition is true and elements from y elsewhere. [[https://docs.scipy.org/doc/numpy/reference/generated/numpy.where.html|numpy.where]]
{{{
>>> x = numpy.array([[5, 4, 3], [2, 1, 0]])
>>> y = numpy.array([[0, 1, 2], [3, 4, 5]])
>>> numpy.where(x > 3, x, y)
[[5 4 2]
 [3 4 5]]
}}}

== Special matrices ==

=== Diagonal matrix ===

Matrix (usually square) in which all entries are zero, except on the main diagonal. Use [[https://docs.scipy.org/doc/numpy-1.10.0/reference/generated/numpy.diag.html | numpy.diag]] to either create a diagonal matrix from a given 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 [[https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.sparse.spdiags.html#scipy.sparse.spdiags|scipy.spare.spdiags]].

=== Identity matrix ===

Special diagonal ''m''*''m'' matrix where all elements on the main diagonal are 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 [[https://docs.scipy.org/doc/numpy/reference/generated/numpy.identity.html|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 [[https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.sparse.identity.html|scipy.sparse.identity]].

=== Triangular matrix ===

A (square) matrix where all elements below (upper triangle) or above (lower triangle) the main diagonal are zero. 
[[https://docs.scipy.org/doc/numpy/reference/generated/numpy.triu.html|numpy.triu]] creates the upper ({{{u}}}), [[https://docs.scipy.org/doc/numpy/reference/generated/numpy.tril.html|numpy.tril]] 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 [[https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.sparse.triu.html|scipy.sparse.triu]] and [[https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.sparse.tril.html|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 [[https://en.wikipedia.org/wiki/Rank_(linear_algebra)| rank]] of ''A''.

Use [[https://docs.scipy.org/doc/numpy/reference/generated/numpy.linalg.svd.html|numpy.linalg.svd]] to do a singular value decomposition for a dense matrix. Use [[https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.svds.html|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)
Uk:
 [[ 0.56475636 -0.30288472]
 [ 0.51457155 -0.59799935]
 [-0.64518709 -0.74206309]] 
Sk:
 [  2.13530566  11.67829513] 
Vk:
 [[-0.52332762 -0.32001209  0.78975975]
 [-0.49726421 -0.63794803 -0.58800563]]
}}}