Types of Matrices

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations

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◇Overview

Square Matrix: If $m=n$ (number of rows equals number of columns). It's called a square matrix of order n. (Handout 1, Page 5)

◇Key Concepts

◇Square Matrix ◇Row Matrix ◇Column Matrix ◇Zero Matrix ($\mathbf{O}$)

◇Details / Theory

◇Square Matrix: If $m=n$ (number of rows equals number of columns). It's called a square matrix of order n. (Handout 1, Page 5)
Example: $\begin{bmatrix} 2 & 1 & 0 \\ -1 & 4 & 2 \\ 10 & 5 & 3 \end{bmatrix}$ is a square matrix of order 3. (Handout 1, Page 5) ◇Row Matrix: If $m=1$ (only one row). Size is $1 \times n$. (Handout 1, Page 5)
Example: $\begin{bmatrix} 1 & 2 & -1 & 4 \end{bmatrix}$ is a $1 \times 4$ row matrix. (Handout 1, Page 5) ◇Column Matrix: If $n=1$ (only one column). Size is $m \times 1$. (Handout 1, Page 5)
Example: $\begin{bmatrix} 1/2 \\ 3 \\ -2/3 \end{bmatrix}$ is a $3 \times 1$ column matrix. (Handout 1, Page 5) ◇Zero Matrix ($\mathbf{O}$): A matrix where all entries are zero ($a_{ij}=0$ for all $i,j$). Often denoted by $O_{m \times n}$ to specify size. (Handout 1, Page 16)
Example: If $A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$
the corresponding zero matrix is
$O=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$ (Handout 1, Page 16)

$$ PN: The zero matrix acts like the number zero in matrix addition: $A+O=A$.

◇Common Pitfalls

Don't confuse matrix type names — row vs. column, upper vs. lower triangular. The type hierarchy matters: identity ⊂ scalar ⊂ diagonal ⊂ triangular.

◇Summary

TL;DR — Matrices are classified by shape (row, column, square) and structure (diagonal, triangular, symmetric, etc.). Knowing the type enables computation shortcuts and determines available operations.

◇Links

Related: - ◇Square Matrix - ◇Row Matrix - ◇Column Matrix - ◇Zero Matrix Source / Reference:

◇References & Navigation

• Previous Topic: ◇Triangular Matrices
• Next Topic: ◇Upper Triangular Matrix
• Related Notes: ◇Main Diagonal, ◇Triangular Matrices, ◇Matrices Detailed