Matrix types

Square Matrix

A matrix where the number of rows(m) equals to the number of columns(n).

Main diagonal: The vector of values along the diagonal of the matrix from the top left to the bottom right. $(a_{11}, a_{22}, a_{33})$

$$Square\hspace{0.1cm}Matrix\hspace{0.1cm}of\hspace{0.1cm}order\hspace{0.1cm}3;\hspace{0.1cm}A=\begin{bmatrix} a_{11} \hspace{0.2cm} a_{12} \hspace{0.2cm} a_{13} \\ a_{21} \hspace{0.2cm} a_{22}\hspace{0.2cm} a_{23} \\ a_{31} \hspace{0.2cm} a_{32}\hspace{0.2cm} a_{33} \end{bmatrix}$$

Symmetric Matrix

A type of square matrix where the top-right triangle is the same as the bottom-left triangle. Note: The axis of symmetry is always the main diagonal.

$$A=\begin{bmatrix} 1 \hspace{0.2cm} 2 \hspace{0.2cm} 3\hspace{0.2cm}4\hspace{0.2cm}5 \\ 2\hspace{0.2cm}1\hspace{0.2cm}2\hspace{0.2cm}3\hspace{0.2cm}4 \\ 3\hspace{0.2cm}2\hspace{0.2cm}1\hspace{0.2cm}2\hspace{0.2cm}3 \\ 4\hspace{0.2cm}3\hspace{0.2cm}2\hspace{0.2cm}1\hspace{0.2cm}2 \\ 5\hspace{0.2cm}4\hspace{0.2cm}3\hspace{0.2cm}2\hspace{0.2cm}1 \end{bmatrix}, A = A^T$$

Triangular Matrix

A type of square matrix that has values in the upper-right or lower-left triangle and filled with zeros in the rest.

$$Upper Triangular Matrix A=\begin{bmatrix} 1 \hspace{0.2cm} 2 \hspace{0.2cm} 3 \\ 0 \hspace{0.2cm} 2\hspace{0.2cm} 3 \\ 0 \hspace{0.2cm} 0\hspace{0.2cm} 3 \end{bmatrix}, Lower Triangular Matrix B=\begin{bmatrix} 1 \hspace{0.2cm} 0 \hspace{0.2cm} 0 \\ 1 \hspace{0.2cm} 2\hspace{0.2cm} 0 \\ 1 \hspace{0.2cm} 2\hspace{0.2cm} 3 \end{bmatrix}$$

from numpy import array
from numpy import tril
from numpy import triu

M = array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])
upper = triu(M)
lower = tril(M)
print(upper)
print(lower)

[[1 2 3]
 [0 2 3]
 [0 0 3]]
 
 [[1 0 0]
 [1 2 0]
 [1 2 3]]

Diagonal Matrix

A matrix where values outside the main diagonal have zero value; often represented as D. Note: A diagonal matrix does not have to be square.

$$D=\begin{bmatrix} 1 \hspace{0.2cm} 0 \hspace{0.2cm} 0 \\ 0 \hspace{0.2cm} 2\hspace{0.2cm} 0 \\ 0 \hspace{0.2cm} 0\hspace{0.2cm} 3 \end{bmatrix}, D=\begin{bmatrix} 1 \hspace{0.2cm} 0 \hspace{0.2cm} 0 \hspace{0.2cm} 0 \\ 0 \hspace{0.2cm} 2\hspace{0.2cm} 0 \hspace{0.2cm} 0 \\ 0 \hspace{0.2cm} 0\hspace{0.2cm} 3 \hspace{0.2cm} 0 \\ 0 \hspace{0.2cm} 0\hspace{0.2cm} 0 \hspace{0.2cm} 4 \\ 0 \hspace{0.2cm} 0\hspace{0.2cm} 0 \hspace{0.2cm} 0 \end{bmatrix}$$

from numpy import array
from numpy import diag
M = array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])

# extract diagonal vector
d = diag(M)
print(d)

# create diagonal matrix from diagonal vector
D = diag(d)
print(D)

[1 2 3]

[[1 0 0]
 [0 2 0]
 [0 0 3]]

Identity Matrix

A square matrix that does not change a vector when multiplied; often represented as 'I' or 'In'.

$$I=\begin{bmatrix} 1 \hspace{0.2cm} 0 \hspace{0.2cm} 0 \\ 0 \hspace{0.2cm} 1\hspace{0.2cm} 0 \\ 0 \hspace{0.2cm} 0\hspace{0.2cm} 1 \end{bmatrix}$$

from numpy import identity
I = identity(3)
print(I)

[[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]

Orthogonal Matrix

Recap: Two vectors are orthogonal when their dot product equals zero, called orthonormal.

Orthogonal matrix is a square matrix whose columns and rows are orthonormal unit vectors; i.e perpendicular and also have a length/magnitude of 1. It is often denoted as 'Q'.

$$Q^T . Q = Q . Q^T = I \hspace{0.5cm}where\hspace{0.1cm}Q^T:Transpose \hspace{0.1cm}of\hspace{0.1cm} Q \newline Q^T = Q^{-1}\hspace{0.5cm}where\hspace{0.1cm}Q^{-1}:Inverse\hspace{0.1cm}of\hspace{0.1cm} Q$$

from numpy import array
from numpy import dot
from numpy import identity
from numpy.linalg import inv

Q = array([[1, 0], [0, -1]])
transpose = Q.T
inverse = inv(Q)
I = identity(2)

# inverse equivalence
print(transpose)
print(inverse)

# identity equivalence
dotproduct = dot(Q,transpose)
print(dotproduct)
print(I)​

[[ 1  0]
 [ 0 -1]]
[[ 1.  0.]
 [-0. -1.]]
[[1 0]
 [0 1]]
[[1. 0.]
 [0. 1.]]

Link: - Introduction to Matrix Types in Linear Algebra for Machine Learning