Pyhton: NumPy Matrix dan Aljabar Linear - Revision history

Onnowpurbo: /* Quadrature */

2023-04-08T00:27:07Z

Quadrature

Onnowpurbo: /* Eigen Value dan Vector */

2023-04-08T00:22:12Z

Eigen Value dan Vector

Onnowpurbo: /* Rank */

2023-04-08T00:10:50Z

Rank

Onnowpurbo: /* Vector sebagai matrix */

2023-04-07T23:48:36Z

Vector sebagai matrix

Onnowpurbo: /* NumPy Matrix */

2023-04-07T23:43:30Z

NumPy Matrix

Onnowpurbo: /* Apa beda antara numpy dot() and inner() */

2023-04-07T23:42:28Z

Apa beda antara numpy dot() and inner()

Onnowpurbo at 23:39, 7 April 2023

2023-04-07T23:39:05Z

Onnowpurbo at 01:54, 4 December 2015

2015-12-04T01:54:31Z

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Onnowpurbo: New page: Sumber: http://www.bogotobogo.com/python/python_numpy_matrix_tutorial.php difference between numpy dot() and inner() What's difference between numpy dot() and inner()? Let's look into ...

2015-12-03T02:39:22Z

New page: Sumber: http://www.bogotobogo.com/python/python_numpy_matrix_tutorial.php difference between numpy dot() and inner() What's difference between numpy dot() and inner()? Let's look into ...

New page

Sumber: http://www.bogotobogo.com/python/python_numpy_matrix_tutorial.php

difference between numpy dot() and inner()

What's difference between numpy dot() and inner()?

Let's look into 2D array as an example:

>>> a=np.array([[1,2],[3,4]])
>>> b=np.array([[11,12],[13,14]])
>>> np.dot(a,b)
array([[37, 40],
[85, 92]])
>>> np.inner(a,b)
array([[35, 41],
[81, 95]])

[1324] [11131214]

With dot():
[1∗11+2∗133∗11+4∗131∗12+2∗143∗12+4∗14] = [37854092]

With inner():
[1∗11+2∗123∗11+4∗121∗13+2∗143∗13+4∗14] = [35814195]

NumPy Matrix

The chapters on NumPy have been using arrays (NumPy Array Basics A and NumPy Array Basics B). However, for certain areas such as linear algebra, we may instead want to use matrix.

>>> import numpy as np
>>> A = np.matrix([[1.,2], [3,4], [5,6]])
>>> A
matrix([[ 1., 2.],
[ 3., 4.],
[ 5., 6.]])

We may also take the Matlab style by giving a string rather than a list:

>>> B = np.matrix("1.,2; 3,4; 5,6")
>>> B
matrix([[ 1., 2.],
[ 3., 4.],
[ 5., 6.]])

A vector as a matrix

Vectors are handled as matrices with one row or one column:

>>> x = np.matrix("10., 20.")
>>> x
matrix([[ 10., 20.]])
>>> x.T
matrix([[ 10.],
[ 20.]])

Here is an example for matrix and vector multiplication:

>>> x = np.matrix("4.;5.")
>>> x
matrix([[ 4.],
[ 5.]])

>>> A = np.matrix([[1.,2], [3,4], [5,6]])
>>> A
matrix([[ 1., 2.],
[ 3., 4.],
[ 5., 6.]])

>>> A*x
matrix([[ 14.],
[ 32.],
[ 50.]])

For vectors, indexing requires two indices:

>>> print x[0,0], x[1,0]
4.0 5.0

Note

Though np.matrix takes a real matrix form and look pleasing, usually, for most of the cases, arrays are good enough.

Rank

>>> import numpy as np
>>> A = np.ones((4,3))
>>> A
array([[ 1., 1., 1.],
[ 1., 1., 1.],
[ 1., 1., 1.],
[ 1., 1., 1.]])
>>> np.rank(A)
2

Note that the rank of the array is not the rank of the matrix in linear algebra (dimension of the column space) but the number of subscripts it takes!

Scalars have rank 0:

>>> x = np.array(10)
>>> x
array(10)
>>> np.rank(x)
0

NumPy supports arrays of any dimension such as rank 3 (2x2x2):

>>> A = np.ones((2,2,2))
>>> A
array([[[ 1., 1.],
[ 1., 1.]],

[[ 1., 1.],
[ 1., 1.]]])
>>> A[1,0,1]
1.0

dot product

>>> A = np.array([[1,2],[3,4]])
>>> A
array([[1, 2],
[3, 4]])
>>> b = np.array([10, 20])
>>> b
array([10, 20])
>>> ans = np.dot(A,b)
>>> ans
array([ 50, 110])

Ax = b : numpy.linalg

Now we want to solve Ax = b:

>>> import numpy as np
>>> from numpy.linalg import solve
>>> A = np.array([[1,2],[3,4]])
>>> A
array([[1, 2],
[3, 4]])
>>> b = np.array([10, 20])
>>> b
>>> x = solve(A,b)
>>> x
array([ 0., 5.])

eigen values and vectors

>>> import numpy as np
>>> from numpy.linalg import eig
>>> A = np.array([[1,2],[3,4]])
>>> eig(A)
(array([-0.37228132, 5.37228132]), array([[-0.82456484, -0.41597356],
[ 0.56576746, -0.90937671]]))

The eig returns two tuples: the first one is the eigen values and the second one is a matrix whose columns are the two eigen vectors.

We can unpack the tuples:

>>> eigen_val, eigen_vec = eig(A)
>>> eigen_val
array([-0.37228132, 5.37228132])
>>> eigen_vec
array([[-0.82456484, -0.41597356],
[ 0.56576746, -0.90937671]])

Quadrature

We want to solve ∫30x4dx=2434:

>>> from scipy.integrate import quad
>>> def f(x):
... return x**4
...
>>> quad(f, 0., 3.)
(48.599999999999994, 5.39568389967826e-13)

The returned tuple indicates (ans, error estimate).

We can get the same answer if we use lambda instead:

>>> quad(lambda x: x**4, 0, 3)
(48.599999999999994, 5.39568389967826e-13)

==Referensi==

* http://www.bogotobogo.com/python/python_numpy_matrix_tutorial.php

@@ Line 52: / Line 52: @@
 ==Vector sebagai matrix==
-Vectors are handled as matrices with one row or one column:
+Vektor ditangani sebagai matriks dengan satu baris atau satu kolom:
-  >>> x = np.matrix("10., 20.")
+  x = np.matrix("10., 20.")
-  >>> x
-Here is an example for matrix and vector multiplication:
+ matrix([[10., 20.]])
-  >>> x = np.matrix("4.;5.")
+  x.T
-For vectors, indexing requires two indices:
+ matrix([[10.],
- >>> print x[0,0], x[1,0]
+Berikut adalah contoh perkalian matriks dan vektor:
-Note
+Untuk vektor, pengindeksan memerlukan dua indeks:
-Though np.matrix takes a real matrix form and look pleasing, usually, for most of the cases, arrays are good enough.
+ print( x[0,0], x[1,0] )
 ==Rank==

@@ Line 36: / Line 36: @@
+  A
   matrix([[1., 2.],
-  matrix([[ 1.,  2.],
+          [3., 4.],
-          [ 3.,  4.],
+          [5., 6.]])
 Kami juga dapat menggunakan gaya Matlab dengan memberikan string bukan list:
@@ Line 46: / Line 45: @@
   B = np.matrix("1.,2; 3,4; 5,6")
+  B
 ==Vector sebagai matrix==

@@ Line 238: / Line 238: @@
 ==Quadrature==
-We want to solve ∫30x4dx=2434:
+Kita ingin menghitung ∫03 x^4 dx=2434:
-  >>> from scipy.integrate import quad
+  from scipy.integrate import quad
-  >>> def f(x):
+  def f(x): return x**4
- ...     return x**4
+  quad(f, 0., 3.)
   (48.599999999999994, 5.39568389967826e-13)
+ (48.599999999999994, 5.39568389967826e-13)
 ==Referensi==
 * http://www.bogotobogo.com/python/python_numpy_matrix_tutorial.php

@@ Line 210: / Line 210: @@
 ==Eigen Value dan Vector==
-  >>> import numpy as np
+  import numpy as np
-  >>> from numpy.linalg import eig
+  from numpy.linalg import eig
-  >>> A = np.array([[1,2],[3,4]])
+  A = np.array([[1,2],[3,4]])
-  >>> eig(A)
+  eig(A)
-We can unpack the tuples:
+ (array([-0.37228132,  5.37228132]),
   array([-0.37228132,  5.37228132])
-  >>> eigen_vec
   array([[-0.82456484, -0.41597356],
          [ 0.56576746, -0.90937671]])
 ==Quadrature==

@@ Line 97: / Line 97: @@
 ==Rank==
-  >>> import numpy as np
+  import numpy as np
-  >>> A = np.ones((4,3))
+  A = np.ones((4,3))
-  >>> A
   array([[ 1.,  1.,  1.],
          [ 1.,  1.,  1.],
          [ 1.,  1.,  1.],
          [ 1.,  1.,  1.]])
-Note that the rank of the array is not the rank of the matrix in linear algebra (dimension of the column space) but the number of subscripts it takes!
+ # np.rank(A)   # rank sudah sudah tidak ada NumPy
-Scalars have rank 0:
+ x = np.array(10)
   array(10)
-  >>> np.rank(x)
-  >>> A = np.ones((2,2,2))
-  >>> A
+NumPy mendukung array dari dimensi apa pun seperti peringkat 3 (2x2x2):
-  array([[[ 1.,  1.],
-         [ 1.,  1.]],
+  A = np.ones((2,2,2))
-         [[ 1.,  1.],
+         [[1., 1.],
-          [ 1.,  1.]]])
+          [1., 1.]]])
- >>> A[1,0,1]
 dot product
-  >>> A = np.array([[1,2],[3,4]])
+  A = np.array([[1,2],[3,4]])
-  >>> A
   array([[1, 2],
          [3, 4]])
-  >>> b = np.array([10, 20])
-  >>> b
+  b = np.array([10, 20])
   array([10, 20])
-  >>> ans = np.dot(A,b)
-  >>> ans
   array([ 50, 110])
@@ Line 148: / Line 185: @@
- Ax = b : numpy.linalg
+  Ax = b : numpy.linalg
-Now we want to solve Ax = b:
+ import numpy as np
   array([[1, 2],
          [3, 4]])
 ==Eigen Value dan Vector==

@@ Line 27: / Line 27: @@
   array([[35, 41],
          [81, 95]])
 ==NumPy Matrix==