7.1 Entry-by-Entry Definition of Matrix

Types: N/A
Examples: 7.3 Examples of Matrix Models, 7.4 Undirected Graphs
Constructions: N/A
Generalizations: N/A

Properties: 7.2 Dimensions of a Matrix, 7.6 Equal Matrices, 18.10 Rank of a Matrix
Sufficiencies: N/A
Questions: N/A

Entry-by-Entry Definition of an

m \times n

Matrix

Let $m, n \in N$ . An $m \times n$ matrix is a rectangular array of $m \cdot n$ real numbers organized into $m$ rows and $n$ columns. We can write the general structure of an $m \times n$ matrix $A$ as:

A = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{matrix}]

Each of the numbers in this array is called an entry, element, or coefficient of the matrix.
Each entry has a row and column index which identifies where in the matrix that coefficient is stored

This is known as the entry-by-entry definition because we literally specified each entry in defining matrix $A$ .

Remark. Each individual entry/element/coefficient of matrix $A$ has three pieces of information:

\begin{aligned} a_{i k} & i \in [m] & k \in [n] \end{aligned}

Row index $i \in {1, 2, \dots, m}$
Column index $k \in {1, 2, \dots, n}$
Real-valued number $a_{i k} \in R$

The value $a_{i k}$ is called the entry/element/coefficient of the matrix.
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Remark. From our definition above, notice that all vectors are matrices but not all matrices are vectors.
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