numpy创建单位矩阵和对角矩阵

在学习linear regression时经常处理的数据一般多是矩阵或者n维向量的数据形式，所以必须对矩阵有一定的认识基础。
numpy中创建单位矩阵借助identity（）函数。更为准确的说，此函数创建的是一个n*n的单位数组，返回值的dtype=array数据形式。其中接受的参数有两个，第一个是n值大小，第二个为数据类型，一般为浮点型。单位数组的概念与单位矩阵相同，主对角线元素为1，其他元素均为零，等同于单位1。而要想得到单位矩阵，只要用mat（）函数将数组转换为矩阵即可。

>>> import numpy as np
>>> help(np.identity)Help on function identity in module numpy:identity(n, dtype=None)Return the identity array.The identity array is a square array with ones onthe main diagonal.Parameters----------n : intNumber of rows (and columns) in `n` x `n` output.dtype : data-type, optionalData-type of the output.  Defaults to ``float``.Returns-------out : ndarray`n` x `n` array with its main diagonal set to one,and all other elements 0.Examples-------->>> np.identity(3)array([[ 1.,  0.,  0.],[ 0.,  1.,  0.],[ 0.,  0.,  1.]])
>>> np.identity(5)array([[1., 0., 0., 0., 0.],[0., 1., 0., 0., 0.],[0., 0., 1., 0., 0.],[0., 0., 0., 1., 0.],[0., 0., 0., 0., 1.]])
>>> A = np.mat(np.identity(5))>>> Amatrix([[1., 0., 0., 0., 0.],[0., 1., 0., 0., 0.],[0., 0., 1., 0., 0.],[0., 0., 0., 1., 0.],[0., 0., 0., 0., 1.]])

矩阵的运算中还经常使用对角阵，numpy中的对角阵用eye（）函数来创建。eye（）函数接受五个参数，返回一个单位数组。第一个和第二个参数N，M分别对应表示创建数组的行数和列数，当然当你只设定一个值时，就默认了N=M。第三个参数k是对角线指数，跟diagonal中的offset参数是一样的，默认值为0，就是主对角线的方向，上三角方向为正，下三角方向为负，可以取-n到+m的范围。第四个参数是dtype，用于指定元素的数据类型，第五个参数是order，用于排序，有‘C’和‘F’两个参数，默认值为‘C’，为行排序，‘F’为列排序。返回值为一个单位数组。

>>> help(np.eye)Help on function eye in module numpy:eye(N, M=None, k=0, dtype=<class 'float'>, order='C')Return a 2-D array with ones on the diagonal and zeros elsewhere.Parameters----------N : intNumber of rows in the output.M : int, optionalNumber of columns in the output. If None, defaults to `N`.k : int, optionalIndex of the diagonal: 0 (the default) refers to the main diagonal,a positive value refers to an upper diagonal, and a negative valueto a lower diagonal.dtype : data-type, optionalData-type of the returned array.order : {'C', 'F'}, optionalWhether the output should be stored in row-major (C-style) orcolumn-major (Fortran-style) order in memory... versionadded:: 1.14.0Returns-------I : ndarray of shape (N,M)An array where all elements are equal to zero, except for the `k`-thdiagonal, whose values are equal to one.See Also--------identity : (almost) equivalent functiondiag : diagonal 2-D array from a 1-D array specified by the user.Examples-------->>> np.eye(2, dtype=int)array([[1, 0],[0, 1]])>>> np.eye(3, k=1)array([[ 0.,  1.,  0.],[ 0.,  0.,  1.],[ 0.,  0.,  0.]])

numpy中的diagonal（）方法可以对n*n的数组和方阵取对角线上的元素，diagonal（）接受三个参数。第一个offset参数是主对角线的方向，默认值为0是主对角线，上三角方向为正，下三角方向为负，可以取-n到+n的范围。第二个参数和第三个参数是在数组大于2维时指定一个2维数组时使用，默认值axis1=0，axis2=1。

>>> help(A.diagonal)Help on built-in function diagonal:diagonal(...) method of numpy.matrix instancea.diagonal(offset=0, axis1=0, axis2=1)Return specified diagonals. In NumPy 1.9 the returned array is aread-only view instead of a copy as in previous NumPy versions.  Ina future version the read-only restriction will be removed.Refer to :func:`numpy.diagonal` for full documentation.See Also--------numpy.diagonal : equivalent function
>>> help(np.diagonal)Help on function diagonal in module numpy:diagonal(a, offset=0, axis1=0, axis2=1)Return specified diagonals.If `a` is 2-D, returns the diagonal of `a` with the given offset,i.e., the collection of elements of the form ``a[i, i+offset]``.  If`a` has more than two dimensions, then the axes specified by `axis1`and `axis2` are used to determine the 2-D sub-array whose diagonal isreturned.  The shape of the resulting array can be determined byremoving `axis1` and `axis2` and appending an index to the right equalto the size of the resulting diagonals.In versions of NumPy prior to 1.7, this function always returned a new,independent array containing a copy of the values in the diagonal.In NumPy 1.7 and 1.8, it continues to return a copy of the diagonal,but depending on this fact is deprecated. Writing to the resultingarray continues to work as it used to, but a FutureWarning is issued.Starting in NumPy 1.9 it returns a read-only view on the original array.Attempting to write to the resulting array will produce an error.In some future release, it will return a read/write view and writing tothe returned array will alter your original array.  The returned arraywill have the same type as the input array.If you don't write to the array returned by this function, then you canjust ignore all of the above.If you depend on the current behavior, then we suggest copying thereturned array explicitly, i.e., use ``np.diagonal(a).copy()`` insteadof just ``np.diagonal(a)``. This will work with both past and futureversions of NumPy.Parameters----------a : array_likeArray from which the diagonals are taken.offset : int, optionalOffset of the diagonal from the main diagonal.  Can be positive ornegative.  Defaults to main diagonal (0).axis1 : int, optionalAxis to be used as the first axis of the 2-D sub-arrays from whichthe diagonals should be taken.  Defaults to first axis (0).axis2 : int, optionalAxis to be used as the second axis of the 2-D sub-arrays fromwhich the diagonals should be taken. Defaults to second axis (1).Returns-------array_of_diagonals : ndarrayIf `a` is 2-D, then a 1-D array containing the diagonal and of thesame type as `a` is returned unless `a` is a `matrix`, in which casea 1-D array rather than a (2-D) `matrix` is returned in order tomaintain backward compatibility.If ``a.ndim > 2``, then the dimensions specified by `axis1` and `axis2`are removed, and a new axis inserted at the end corresponding to thediagonal.Raises------ValueErrorIf the dimension of `a` is less than 2.See Also--------diag : MATLAB work-a-like for 1-D and 2-D arrays.diagflat : Create diagonal arrays.trace : Sum along diagonals.Examples-------->>> a = np.arange(4).reshape(2,2)>>> aarray([[0, 1],[2, 3]])>>> a.diagonal()array([0, 3])>>> a.diagonal(1)array([1])A 3-D example:>>> a = np.arange(8).reshape(2,2,2); aarray([[[0, 1],[2, 3]],[[4, 5],[6, 7]]])>>> a.diagonal(0, # Main diagonals of two arrays created by skipping...            0, # across the outer(left)-most axis last and...            1) # the "middle" (row) axis first.array([[0, 6],[1, 7]])The sub-arrays whose main diagonals we just obtained; note that eachcorresponds to fixing the right-most (column) axis, and that thediagonals are "packed" in rows.>>> a[:,:,0] # main diagonal is [0 6]array([[0, 2],[4, 6]])>>> a[:,:,1] # main diagonal is [1 7]array([[1, 3],[5, 7]])
>>> A = np.random.randint(low=5, high=30, size=(5, 5))>>> Aarray([[25, 15, 26,  6, 22],[27, 14, 22, 16, 21],[22, 17, 10, 14, 25],[11,  9, 27, 20,  6],[24, 19, 19, 26, 14]])
>>> A.diagonal()array([25, 14, 10, 20, 14])
>>> A.diagonal(offset=1)array([15, 22, 14,  6])
>>> A.diagonal(offset=-2)array([22,  9, 19])

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