9.13 Transpose of a Matrix

Types: N/A
Examples: N/A
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Generalizations: N/A

Properties: 9.14 Algebraic Properties of Matrix Transposes
Sufficiencies: N/A
Questions: N/A

Transpose of a Matrix

Let matrix $A \in R^{m \times n}$ . The transpose of $A$ , denoted by $A^{T} \in R^{n \times m}$ , is formed by turning all columns of $A$ into rows of $A^{T}$ and vice versa.

\begin{aligned} A = {[\begin{array}{c} a & b & c \\ d & e & f \end{array}]}_{2 \times 3} & A^{T} = {[\begin{array}{c} a & d \\ b & e \\ c & f \end{array}]}_{3 \times 2} \end{aligned}

Remark. A nice way to visualize this is to imagine the matrix being "reflected" over a diagonal.
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Transpose of a Matrix by Columns

Form $A^{T} \in R^{m \times n}$ by columns. Let $k \in [m]$ .

Using 8.11 Colon Notation, we can write our columns of matrix $A^{T}$ .

[A^{T} (:, k)]_{n \times 1} = {Column}_{k} (A^{T}) = [{Row}_{k} (A)]^{T} = [A (k, :)]^{T}

$k$ is the identifier we are using for columns. To form the transpose of $A$ by columns, we simply rewrite each column as a row.

Transpose of a Matrix by Rows

Form $A^{T} \in R^{m \times n}$ by rows. Let $i \in [n]$ .

Again, we can use colon notation.

A^{T} (i, :) = {Row}_{i} (A^{T}) = [{Column}_{i} (A)]^{T} = [A (:, i)]^{T}

Transpose of a Matrix Entry-by-Entry

Form $A^{T} \in R^{m \times n}$ entry-by-entry. Let $k \in [m], i \in [n]$ .

A^{T} (i, k) = {Entry}_{i k} (A^{T}) = {Entry}_{k i} (A)