python矩阵对角化_numpy创建单位矩阵和对角矩阵的实例
在學習linear regression時經常處理的數據一般多是矩陣或者n維向量的數據形式，所以必須對矩陣有一定的認識基礎。
numpy中創建單位矩陣借助identity()函數。更為準確的說，此函數創建的是一個n*n的單位數組，返回值的dtype=array數據形式。其中接受的參數有兩個，第一個是n值大小，第二個為數據類型，一般為浮點型。單位數組的概念與單位矩陣相同，主對角線元素為1，其他元素均為零，等同于單位1。而要想得到單位矩陣，只要用mat()函數將數組轉換為矩陣即可。
>>> 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 on
the main diagonal.
Parameters
----------
n : int
Number of rows (and columns) in `n` x `n` output.
dtype : data-type, optional
Data-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))
>>> A
matrix([[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.]])
矩陣的運算中還經常使用對角陣，numpy中的對角陣用eye()函數來創建。eye()函數接受五個參數，返回一個單位數組。第一個和第二個參數N，M分別對應表示創建數組的行數和列數，當然當你只設定一個值時，就默認了N=M。第三個參數k是對角線指數，跟diagonal中的offset參數是一樣的，默認值為0，就是主對角線的方向，上三角方向為正，下三角方向為負，可以取-n到+m的范圍。第四個參數是dtype，用于指定元素的數據類型，第五個參數是order，用于排序，有‘C'和‘F'兩個參數，默認值為‘C'，為行排序，‘F'為列排序。返回值為一個單位數組。
>>> help(np.eye)
Help on function eye in module numpy:
eye(N, M=None, k=0, dtype=, order='C')
Return a 2-D array with ones on the diagonal and zeros elsewhere.
Parameters
----------
N : int
Number of rows in the output.
M : int, optional
Number of columns in the output. If None, defaults to `N`.
k : int, optional
Index of the diagonal: 0 (the default) refers to the main diagonal,
a positive value refers to an upper diagonal, and a negative value
to a lower diagonal.
dtype : data-type, optional
Data-type of the returned array.
order : {'C', 'F'}, optional
Whether the output should be stored in row-major (C-style) or
column-major (Fortran-style) order in memory.
.. versionadded:: 1.14.0
Returns
-------
I : ndarray of shape (N,M)
An array where all elements are equal to zero, except for the `k`-th
diagonal, whose values are equal to one.
See Also
--------
identity : (almost) equivalent function
diag : 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()方法可以對n*n的數組和方陣取對角線上的元素，diagonal()接受三個參數。第一個offset參數是主對角線的方向，默認值為0是主對角線，上三角方向為正，下三角方向為負，可以取-n到+n的范圍。第二個參數和第三個參數是在數組大于2維時指定一個2維數組時使用，默認值axis1=0，axis2=1。
>>> help(A.diagonal)
Help on built-in function diagonal:
diagonal(...) method of numpy.matrix instance
a.diagonal(offset=0, axis1=0, axis2=1)
Return specified diagonals. In NumPy 1.9 the returned array is a
read-only view instead of a copy as in previous NumPy versions. In
a 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 is
returned. The shape of the resulting array can be determined by
removing `axis1` and `axis2` and appending an index to the right equal
to 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 resulting
array 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 to
the returned array will alter your original array. The returned array
will have the same type as the input array.
If you don't write to the array returned by this function, then you can
just ignore all of the above.
If you depend on the current behavior, then we suggest copying the
returned array explicitly, i.e., use ``np.diagonal(a).copy()`` instead
of just ``np.diagonal(a)``. This will work with both past and future
versions of NumPy.
Parameters
----------
a : array_like
Array from which the diagonals are taken.
offset : int, optional
Offset of the diagonal from the main diagonal. Can be positive or
negative. Defaults to main diagonal (0).
axis1 : int, optional
Axis to be used as the first axis of the 2-D sub-arrays from which
the diagonals should be taken. Defaults to first axis (0).
axis2 : int, optional
Axis to be used as the second axis of the 2-D sub-arrays from
which the diagonals should be taken. Defaults to second axis (1).
Returns
-------
array_of_diagonals : ndarray
If `a` is 2-D, then a 1-D array containing the diagonal and of the
same type as `a` is returned unless `a` is a `matrix`, in which case
a 1-D array rather than a (2-D) `matrix` is returned in order to
maintain 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 the
diagonal.
Raises
------
ValueError
If 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)
>>> a
array([[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); a
array([[[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 each
corresponds to fixing the right-most (column) axis, and that the
diagonals 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))
>>> A
array([[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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