12 Matrix & Array

• Generate p-dimensional array

• Matrix is the 2-dimensional array

• In Python, we need to load numpy library

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12.1 R

X = matrix(c(1, 2, 3, 4), nrow = 2, ncol = 2)

Y = matrix(c(5, 6, 7, 8), nrow = 2, ncol = 2)

Function	Explanation	Example.
dim	Dimension of a matrix (row, col)	dim(X)
nrow	Number of rows of a matrix	nrow(X)
ncol	Number of columns of a matrix	ncol(X)
rbind	Row bind two matrices	rbind(X, Y)
cbind	Column bind two matrices	cbind(X, Y)
rowSums	Rowwise sum of elements of a matrix	rowSums(X)
colSums	Columnwise sum of elements of a matrix	colSums(X)
rank	Rank of a matrix	rank(X)
+	Addition of two matrices	X + Y
-	Subtraction of two matrices	X - Y
*	Elementwise multiplication of two matrices	X * Y
%*%	Multiplication of two matrices	X %*% Y
outer	Outer product of two matrices	outer(X, Y)
cross	Cross product of two matrices	cross(X, Y)
t	Tranpose of a matrix	t(X)
det	Determinant of a matrix	det(X)
solve	Inverse of a matrix	solve(X)

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12.2 R Matrix Indexing

• R Matrix is indexed by row and column

• MatrixObj[row, col]

• Example: MatrixObj[2, 1] gets the element at the second row and first column

• Keeping it blank in the first (second) position will return all rows (all columns)

• Example: MatrixObj[, 1] returns all rows of the first column

• Example: MatrixObj[2, ] returns all columns of the second row

• Remember the R indexing starts at 1

12.3 Python

import numpy as np

X = np.array([[1,2], [3,4]])

Y = np.array([[5,6], [7,8]])

Method	Explanation	Example.
ndim	Dimension of an array	X.ndim
shape	Shape of a matrix or array (row, col)	X.shape
shape	Number of rows of a matrix	X.shape[0]
shape	Number of columns of a matrix	X.shape[1]
vstack	Row bind two matrices	np.vstack((X, Y))
hstack	Column bind two matrices	np.hstack((X, Y))
sum, axis = 0	Rowwise sum of elements of a matrix	X.sum(axis = 0); np.sum(X, axis = 0)
sum, axis = 1	Columnwise sum of elements of a matrix	X.sum(axis = 1); np.sum(X, axis = 1)
rank	Rank of a matrix	np.linalg.matrix_rank(X)
+	Addition of two matrices	X + Y
-	Subtraction of two matrices	X - Y
*	Elementwise multiplication of two matrices	X * Y; np.multiply(X, Y)
%*%	Multiplication of two matrices	np.matmul(X, Y)
outer	Outer product of two matrices	np.outer(X, Y)
cross	Cross product of two matrices	np.cross(X, Y)
t	Tranpose of a matrix	np.transpose(X)
det	Determinant of a matrix	np.linalg.det(X)
solve	Inverse of a matrix	np.linalg.inv(X)

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12.4 Python Matrix Indexing

• Indexing NumPy matrices is fundamentally the same as Python list indexing, although NumPy array can be n-dimensional.

• Slicing and indexing on matrix is same as lists:

MatrixObj[start:stop:step]

• Indexing in one dimension will default to the first dimension (i.e. row):

X[0] = np.array([1,2])

• Specifying the second dimension index will return an element of the matrix:

X[0,1] = 2 or X[0][1] = 2

• To specifiy all elements of one dimension use :.

X[:, 1] = np.array([2,4])

• In the case of ndim > 2 the ellipsis ... inserts as many full slices : to extend the multi-dimensional slice to all dimensions.

X[:,:,:,0] is equivalent to X[...,0]

• Remember the Python indexing starts at 0