22 \begin{center} |
22 \begin{center} |
23 \LARGE{Matrices and Least Square Fit}\\ |
23 \LARGE{Matrices and Least Square Fit}\\ |
24 \large{FOSSEE} |
24 \large{FOSSEE} |
25 \end{center} |
25 \end{center} |
26 \section{Matrices} |
26 \section{Matrices} |
27 Inputting a Matrix |
27 \subsection{Basics} |
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28 Matrix Creation\\ |
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29 \typ{In []: C = array([[1,1,2], [2,4,1], [-1,3,7]])}\\ |
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30 It creates C matrix of shape 3x3\\ |
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31 Shape is dimenions of given array. |
28 \begin{lstlisting} |
32 \begin{lstlisting} |
29 In []: C = array([[1,1,2], |
33 In []: C.shape |
30 [2,4,1], |
34 Out[]: (3, 3) |
31 [-1,3,7]]) |
35 In []: shape([[1,2],[4,5],[3,0]]) |
32 In []: B = ones_like(C) |
36 Out[]: (3, 2) |
33 In []: A = ones((3,2)) |
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34 In []: I = identity(3) |
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35 \end{lstlisting} |
37 \end{lstlisting} |
36 Accessing Elements |
38 \typ{In []: B = ones_like(C)} \\ |
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39 B would be array of ones with the same shape and type as C.\\ |
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40 \typ{In []: A = ones((3,2))} \\ |
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41 A would be new array of given shape(arguments), filled with ones.\\ |
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42 \typ{In []: I = identity(3)}\\ |
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43 I would be identity matrix of shape 3x3 |
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44 |
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45 \subsection{Accessing Elements} |
37 \begin{lstlisting} |
46 \begin{lstlisting} |
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47 In []: C |
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48 Out[]: |
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49 array([[ 1, 1, 2], |
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50 [ 2, 4, 1], |
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51 [-1, 3, 7]]) |
38 In []: C[1,2] |
52 In []: C[1,2] |
39 Out[]: 1 |
53 Out[]: 1 |
40 |
54 \end{lstlisting} |
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55 Two indexes seperated by ',' specifies row, column. So \kwrd{C[1,2]} gets third element of second row(indices starts from 0). |
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56 \newpage |
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57 \begin{lstlisting} |
41 In []: C[1] |
58 In []: C[1] |
42 Out[]: array([2, 4, 1]) |
59 Out[]: array([2, 4, 1]) |
43 \end{lstlisting} |
60 \end{lstlisting} |
44 |
61 Single index implies complete row. |
45 Changing elements |
62 \subsection{Changing elements} |
46 \begin{lstlisting} |
63 \begin{lstlisting} |
47 In []: C[1,1] = -2 |
64 In []: C[1,1] = -2 |
48 In []: C |
65 In []: C |
49 Out[]: |
66 Out[]: |
50 array([[ 1, 1, 2], |
67 array([[ 1, 1, 2], |
57 array([[ 1, 1, 2], |
74 array([[ 1, 1, 2], |
58 [ 0, 0, 0], |
75 [ 0, 0, 0], |
59 [-1, 3, 7]]) |
76 [-1, 3, 7]]) |
60 \end{lstlisting} |
77 \end{lstlisting} |
61 |
78 |
62 Slicing |
79 \subsection{Slicing} |
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80 Accessing rows with Matricies is straightforward. But If one wants to access particular Column, or want a sub-matrix, Slicing is the way to go. |
63 \begin{lstlisting} |
81 \begin{lstlisting} |
64 In []: C[:,1] |
82 In []: C[:,1] |
65 Out[]: array([1, 0, 3]) |
83 Out[]: array([1, 0, 3]) |
66 |
84 \end{lstlisting} |
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85 First index(:) specifies row(':' implies all the rows) and second index(1) specifies column(second column). |
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86 \begin{lstlisting} |
67 In []: C[1,:] |
87 In []: C[1,:] |
68 Out[]: array([0, 0, 0]) |
88 Out[]: array([0, 0, 0]) |
69 |
89 \end{lstlisting} |
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90 Here we get second row(1), all columns(':') of C matrix. |
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91 \newpage |
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92 \begin{lstlisting} |
70 In []: C[0:2,:] |
93 In []: C[0:2,:] |
71 Out[]: |
94 Out[]: |
72 array([[1, 1, 2], |
95 array([[1, 1, 2], |
73 [0, 0, 0]]) |
96 [0, 0, 0]]) |
74 |
97 \end{lstlisting} |
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98 Result is sub-matrix with first and second row(endpoint is excluded), and all columns from C. |
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99 \begin{lstlisting} |
75 In []: C[1:3,:] |
100 In []: C[1:3,:] |
76 Out[]: |
101 Out[]: |
77 array([[ 0, 0, 0], |
102 array([[ 0, 0, 0], |
78 [-1, 3, 7]]) |
103 [-1, 3, 7]]) |
79 |
104 |
80 In []: C[:2,:] |
105 In []: C[:2,:] |
81 Out[]: |
106 Out[]: |
82 array([[1, 1, 2], |
107 array([[1, 1, 2], |
83 [0, 0, 0]]) |
108 [0, 0, 0]]) |
84 |
109 \end{lstlisting} |
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110 \typ{':2'} => start from first row, till and excluding third row. |
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111 \begin{lstlisting} |
85 In []: C[1:,:] |
112 In []: C[1:,:] |
86 Out[]: |
113 Out[]: |
87 array([[ 0, 0, 0], |
114 array([[ 0, 0, 0], |
88 [-1, 3, 7]]) |
115 [-1, 3, 7]]) |
89 |
116 |
90 In []: C[1:,:2] |
117 In []: C[1:,:2] |
91 Out[]: |
118 Out[]: |
92 array([[ 0, 0], |
119 array([[ 0, 0], |
93 [-1, 3]]) |
120 [-1, 3]]) |
94 \end{lstlisting} |
121 \end{lstlisting} |
95 |
122 \typ{'1:'} => Start from second row, till last row\\ |
96 Striding |
123 \typ{':2'} => Start from first column, till and excluding third column. |
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124 \subsection{Striding} |
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125 Often apart from submatrix, one needs to get some mechanism to jump a step. For example, how can we have all alternate rows of a Matrix. \\ |
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126 Following method will return Matrix with alternate rows. |
97 \begin{lstlisting} |
127 \begin{lstlisting} |
98 In []: C[::2,:] |
128 In []: C[::2,:] |
99 Out[]: |
129 Out[]: |
100 array([[ 1, 1, 2], |
130 array([[ 1, 1, 2], |
101 [-1, 3, 7]]) |
131 [-1, 3, 7]]) |
102 |
132 \end{lstlisting} |
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133 \typ{C[startR:stopR:stepR,startC:stopC:stepC]} => Syntax of mentioning starting index, ending index, and step to jump.\\ |
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134 In above mentioned case, \typ{'::2'} means, start from first row, till last row(both are blank), with step of 2, that is, skipping alternate row. After first row, C[startR], next row would be C[startR+stepR] and so on. |
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135 \begin{lstlisting} |
103 In []: C[:,::2] |
136 In []: C[:,::2] |
104 Out[]: |
137 Out[]: |
105 xarray([[ 1, 2], |
138 xarray([[ 1, 2], |
106 [ 0, 0], |
139 [ 0, 0], |
107 [-1, 7]]) |
140 [-1, 7]]) |
108 |
141 \end{lstlisting} |
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142 Same as above, just that here we get matrix with each alternate column and all rows. |
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143 \begin{lstlisting} |
109 In []: C[::2,::2] |
144 In []: C[::2,::2] |
110 Out[]: |
145 Out[]: |
111 array([[ 1, 2], |
146 array([[ 1, 2], |
112 [-1, 7]]) |
147 [-1, 7]]) |
113 \end{lstlisting} |
148 \end{lstlisting} |
114 |
149 |
115 Matrix Operations |
150 \Section{Matrix Operations} |
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151 For a Matrix A and B of equal shapes. |
116 \begin{lstlisting} |
152 \begin{lstlisting} |
117 In []: A.T # Transpose |
153 In []: A.T # Transpose |
118 In []: sum(A) # Sum of all elements |
154 In []: sum(A) # Sum of all elements |
119 In []: A+B # Addition |
155 In []: A+B # Addition |
120 In []: A*B # Product |
156 In []: A*B # Element wise product |
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157 In []: dot(A,b) #Matrix multiplication |
121 In []: inv(A) # Inverse |
158 In []: inv(A) # Inverse |
122 In []: det(A) # Determinant |
159 In []: det(A) # Determinant |
123 \end{lstlisting} |
160 \end{lstlisting} |
124 |
161 |
125 Eigen Values and Eigen Vectors |
162 Eigen Values and Eigen Vectors |