types of matrices

Topics

• matrices

• Square matrix: number or rows = number of columns
  • Has a principal diagonal
• Row and Column matrix: Single row or single column only
• Singleton matrix: Single entry/element only
• Zero/Null matrix: All entries are 0
• Diagonal matrix: All entries apart from principal diagonal are 0
  • Has to be a square matrix
• Scalar matrix: Diagonal matrix with all diagonal elements being same
• Unit/Identity matrix: Diagonal matrix with all diagonal elements being 1
• Triangular matrix: Square matrix in which elements above/below the principal diagonal are 0
  • Upper/Lower triangular matrix, depending on whether non-zero entries are above/below diagonal
• Equivalence matrix: Matrices with same order/dimensions
• Equal matrix: Matrices with same corresponding entries
• Transpose of a matrix: Move rows to columns and vice versa
• Symmetric matrix: Matrix and its transpose are equal
• Skew symmetric matrix: Matrix and negative of its transpose are equal
• Orthogonal matrix: Square matrix Q whose rows and cols are orthonormal unit vectors
  • orthonormal unit vectors: perpendicular and have a length or magnitude of 1
  • $Q{Q}^T=Q^{T}Q=I⟹Q^{-1}=Q^T$

import numpy as np
 
# Square matrix
square = np.array([[1, 2], [3, 4]])
 
# Row matrix
row = np.array([[1, 2, 3]])
 
# Column matrix
col = np.array([[1], [2], [3]])
 
# Singleton matrix
singleton = np.array([5])
 
# Zero matrix
zero = np.zeros((2, 2))
 
# Diagonal matrix
diagonal = np.diag([1, 2, 3])
 
# Scalar matrix
scalar = 5 * np.eye(3)
 
# Identity matrix
identity = np.eye(3)
 
# Upper triangular matrix
upper = np.triu([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
 
# Lower triangular matrix
lower = np.tril([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
 
# Symmetric matrix
symmetric = np.array([[1, 2], [2, 1]])
 
# Skew-symmetric matrix
skew_symmetric = np.array([[0, 2], [-2, 0]])
 
# Orthogonal matrix Q: (Q.T.dot(Q) == np.eye(2))
orthogonal = np.array([[1, 0], [0, -1]])

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