8.1 Matrix Anatomy

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 7.1 Entry-by-Entry Definition of Matrix

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Matrix Anatomy

For $m, n \in N$ , a rectangular matrix $A \in R^{m \times n}$ is an array of real numbers organized into $m$ rows and $n$ columns. Because the row dimension $m$ and the column dimension $n$ are both natural numbers, matrices are categorized as three types:

Tall and narrow if $m > n$ . All tall and narrow matrices have more rows than columns.

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

Square if $m = n$ (not necessarily nonsingular). All square matrices have the same number of rows as columns.

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

Short and wide if $m < n$ . All short and wide matrices have less rows than columns.

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

Remark. The anatomy of matrices can be used to help identify what 00 Major Problems in Applied Linear Algebra the problem may be.
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