In numpy arrays, dimensionality refers to the number of axes needed to index it, not the dimensionality of any geometrical space. For example, you can describe the locations of points in 3D space with a 2D array:
array([ [0, 0, 0], [1, 2, 3], [2, 2, 2], [9, 9, 9] ])
You can play around with this, and see the number of dimensions and shape of an array like so:
In[262]: a = np.arange(9)
In[263]: a
Out[263]: array([0, 1, 2, 3, 4, 5, 6, 7, 8])
In[264]: a.ndim # number of dimensions
Out[264]: 1
In[265]: a.shape
Out[265]: (9, )
In[266]: b = np.array([
[0, 0, 0],
[1, 2, 3],
[2, 2, 2],
[9, 9, 9]
])
In[267]: b
Out[267]:
array([
[0, 0, 0],
[1, 2, 3],
[2, 2, 2],
[9, 9, 9]
])
In[268]: b.ndim
Out[268]: 2
In[269]: b.shape
Out[269]: (4, 3)Arrays can have many dimensions, but they become hard to visualize above two or three:
In[276]: c = np.random.rand(2, 2, 3, 4)
In[277]: c
Out[277]:
array([
[
[
[0.33018579, 0.98074944, 0.25744133, 0.62154557],
[0.70959511, 0.01784769, 0.01955593, 0.30062579],
[0.83634557, 0.94636324, 0.88823617, 0.8997527]
],
[
[0.4020885, 0.94229555, 0.309992, 0.7237458],
[0.45036185, 0.51943908, 0.23432001, 0.05226692],
[0.03170345, 0.91317231, 0.11720796, 0.31895275]
]
],
[
[
[0.47801989, 0.02922993, 0.12118226, 0.94488471],
[0.65439109, 0.77199972, 0.67024853, 0.27761443],
[0.31602327, 0.42678546, 0.98878701, 0.46164756]
],
[
[0.31585844, 0.80167337, 0.17401188, 0.61161196],
[0.74908902, 0.45300247, 0.68023488, 0.79672751],
[0.23597218, 0.78416727, 0.56036792, 0.55973686]
]
]
])
In[278]: c.ndim
Out[278]: 4
In[279]: c.shape
Out[279]: (2, 2, 3, 4)In Numpy, dimension, axis/axes, shape are related and sometimes similar concepts:
In[1]: import numpy as np
In[2]: a = np.array([
[1, 2],
[3, 4]
])In Numpy dimensions are called axes. The number of axes is rank.
In[3]: a.ndim # num of dimensions / axes, * Mathematics definition of dimension *
Out[3]: 2the nth coordinate to index an array in Numpy. And multidimensional arrays can have one index per axis.
In[4]: a[1, 0] # to index `a`, we specific 1 at the first axis and 0 at the second axis.
Out[4]: 3 # which results in 3(locate at the row 1 and column 0, 0 - based index)You can also use axis parameter in group operations, in case of axis=0 Numpy performs the action on elements of each column, and if axis=1, it performs the action on rows.
test = np.arange(0, 9).reshape(3, 3)
Out[3]:
array([
[0, 1, 2],
[3, 4, 5],
[6, 7, 8]
])
test.sum(axis = 0)
Out[5]: array([9, 12, 15])
test.sum(axis = 1)
Out[6]: array([3, 12, 21])In order to understand the dimensions and axes, it is important to understand tensors and its rank. A vector is a rank-1 tensor a matrix is a rank-2 tensor and so on and so forth. Consider the following:
x = np.array([0, 3, 4, 5, 8])
Axes are defined for arrays with more than one dimension. A 2-dimensional array has two corresponding axes: the first running vertically downwards across rows (axis 0), and the second running horizontally across columns (axis 1).,A way to represent items in a N-dimensional array in the 1-dimensional computer memory. In row-major order, the rightmost index “varies the fastest”: for example the array:,A way to represent items in a N-dimensional array in the 1-dimensional computer memory. In column-major order, the leftmost index “varies the fastest”: for example the array:,A 2-dimensional ndarray that preserves its two-dimensional nature throughout operations. It has certain special operations, such as * (matrix multiplication) and ** (matrix power), defined:
>>> x = np.arange(12).reshape((3, 4))
>>>
x
array([
[0, 1, 2, 3],
[4, 5, 6, 7],
[8, 9, 10, 11]
])
>>>
x.sum(axis = 1)
array([6, 22, 38])>>> x = np.array([1, 2, 3], float)
>>>
x
array([1., 2., 3.])
>>>
x.dtype # floating point number, 64 bits of memory per element
dtype('float64')
# More complicated data type: each array element is a combination of
# and integer and a floating point number >>>
np.array([(1, 2.0), (3, 4.0)], dtype = [('x', int), ('y', float)])
array([(1, 2.0), (3, 4.0)],
dtype = [('x', '<i4'), ('y', '<f8')])>>> x = np.array([1, 2, 3]) >>> x.shape(3, )
>>> x = np.array([1, 2]) >>>
y = np.array([
[3],
[4]
])
>>>
x
array([1, 2])
>>>
y
array([
[3],
[4]
])
>>>
x + y
array([
[4, 5],
[5, 6]
])[ [1, 2, 3], [4, 5, 6] ]
[1, 4, 2, 5, 3, 6]Last Updated : 30 May, 2021,GATE CS 2021 Syllabus
Output :
[ [1 2 3] [11 22 33] [4 5 6] [8 9 10] [20 30 40] ] Sum array - wise: 204