#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 == TODO (Raghu): examples of dot product (dense * dense, dense * sparse, sparse * sparse), usage of both matrix.dot() and *, 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 == === 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]] }}}