博客園LaTex的測試，附帶開啓方法

開啓方法

內容部分先不用管，我這邊只是作個測試，過幾天會講這些的，先忽略

---

 

 

 

矩陣系列¶

1.建立特殊矩陣¶

1.1.建立全爲0的矩陣¶

np.zeros(tuple)

$$\begin{bmatrix} 0&0&0 \\ 0&0&0 \\ 0&0&0 \end{bmatrix}$$

In [1]:

import numpy as np

In [2]:

help(np.zeros)

 

Help on built-in function zeros in module numpy.core.multiarray:

zeros(...)
    zeros(shape, dtype=float, order='C')
    
    Return a new array of given shape and type, filled with zeros.
    
    Parameters
    ----------
    shape : int or sequence of ints
        Shape of the new array, e.g., ``(2, 3)`` or ``2``.
    dtype : data-type, optional
        The desired data-type for the array, e.g., `numpy.int8`.  Default is
        `numpy.float64`.
    order : {'C', 'F'}, optional
        Whether to store multidimensional data in C- or Fortran-contiguous
        (row- or column-wise) order in memory.
    
    Returns
    -------
    out : ndarray
        Array of zeros with the given shape, dtype, and order.
    
    See Also
    --------
    zeros_like : Return an array of zeros with shape and type of input.
    ones_like : Return an array of ones with shape and type of input.
    empty_like : Return an empty array with shape and type of input.
    ones : Return a new array setting values to one.
    empty : Return a new uninitialized array.
    
    Examples
    --------
    >>> np.zeros(5)
    array([ 0.,  0.,  0.,  0.,  0.])
    
    >>> np.zeros((5,), dtype=int)
    array([0, 0, 0, 0, 0])
    
    >>> np.zeros((2, 1))
    array([[ 0.],
           [ 0.]])
    
    >>> s = (2,2)
    >>> np.zeros(s)
    array([[ 0.,  0.],
           [ 0.,  0.]])
    
    >>> np.zeros((2,), dtype=[('x', 'i4'), ('y', 'i4')]) # custom dtype
    array([(0, 0), (0, 0)],
          dtype=[('x', '<i4'), ('y', '<i4')])

In [3]:

# 一維
np.zeros(5) # 完整寫法：np.zeros((5,))

Out[3]:

array([0., 0., 0., 0., 0.])

In [4]:

# 能夠指定類型
np.zeros(5,dtype=int)

Out[4]:

array([0, 0, 0, 0, 0])

In [5]:

# 二維
np.zeros((2,5))

Out[5]:

array([[0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0.]])

In [6]:

# 三維 ==> 能夠這麼理解，2個2*5（2行5列）的矩陣
np.zeros((2,2,5))

Out[6]:

array([[[0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 0.]],

       [[0., 0., 0., 0., 0.],
        [0., 0., 0., 0., 0.]]])

In [7]:

################### 擴展部分 ########################

In [8]:

# 建議用元組，官方文檔都是元組，並且shape返回類型就是元組
array1 = np.zeros([2,3])
print(array1)
type(array1)
print(array1.shape) # shape返回類型就是元組

 

[[0. 0. 0.]
 [0. 0. 0.]]
(2, 3)

 

1.2.建立全爲1的矩陣¶

np.ones(tuple) 用法和np.zeros(tuple)差很少

$$\begin{bmatrix} 1&1&1 \\ 1&1&1 \\ 1&1&1 \end{bmatrix}$$

In [9]:

help(np.ones)

 

Help on function ones in module numpy.core.numeric:

ones(shape, dtype=None, order='C')
    Return a new array of given shape and type, filled with ones.
    
    Parameters
    ----------
    shape : int or sequence of ints
        Shape of the new array, e.g., ``(2, 3)`` or ``2``.
    dtype : data-type, optional
        The desired data-type for the array, e.g., `numpy.int8`.  Default is
        `numpy.float64`.
    order : {'C', 'F'}, optional
        Whether to store multidimensional data in C- or Fortran-contiguous
        (row- or column-wise) order in memory.
    
    Returns
    -------
    out : ndarray
        Array of ones with the given shape, dtype, and order.
    
    See Also
    --------
    zeros, ones_like
    
    Examples
    --------
    >>> np.ones(5)
    array([ 1.,  1.,  1.,  1.,  1.])
    
    >>> np.ones((5,), dtype=int)
    array([1, 1, 1, 1, 1])
    
    >>> np.ones((2, 1))
    array([[ 1.],
           [ 1.]])
    
    >>> s = (2,2)
    >>> np.ones(s)
    array([[ 1.,  1.],
           [ 1.,  1.]])

In [10]:

# 一維
np.ones(5) # 完整寫法 np.ones((5,))

Out[10]:

array([1., 1., 1., 1., 1.])

In [11]:

# 能夠指定類型
np.ones(5,dtype=int)

Out[11]:

array([1, 1, 1, 1, 1])

In [12]:

# 二維，傳一個shape元組
np.ones((2,5))

Out[12]:

array([[1., 1., 1., 1., 1.],
       [1., 1., 1., 1., 1.]])

In [13]:

# 三維 能夠理解爲兩個二維數組
np.ones((2,2,5))

Out[13]:

array([[[1., 1., 1., 1., 1.],
        [1., 1., 1., 1., 1.]],

       [[1., 1., 1., 1., 1.],
        [1., 1., 1., 1., 1.]]])

 

1.3.單位矩陣¶

先普及一個數學基礎：任何矩陣 x 單位矩陣 都等於其自己

單位矩陣是個方陣，從左上角到右下角的對角線（稱爲主對角線）上的元素均爲1。其餘全都爲0，eg:

$$\begin{bmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix}$$

用 np.eye() 來定義(eye：眼睛)

擴展：np.eye(rows,columns=rows)

In [14]:

help(np.eye)

 

Help on function eye in module numpy.lib.twodim_base:

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 : 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.]])

In [15]:

# 定義一個2行的單位矩陣（列默認和行一致）
np.eye(2)

Out[15]:

array([[1., 0.],
       [0., 1.]])

In [16]:

np.eye(3,dtype=int)

Out[16]:

array([[1, 0, 0],
       [0, 1, 0],
       [0, 0, 1]])

In [17]:

# 定義一個5行5列的單位矩陣
np.eye(5)

Out[17]:

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.]])