Textbook
Transposing $A\in\mathbb R^{m\times n}$ produces $A^{\mathsf T}\in\mathbb R^{n\times m}$ with $(A^{\mathsf T})_{ji}=A_{ij}$. Rows become columns and columns become rows, preserving the entries. Transposing twice restores the original matrix.
A column of three coordinates has shape $3\times1$; its transpose has shape $1\times3$. The number of coordinates stays the same, but compatible matrix products depend on the new shape. The superscript $\mathsf T$ rearranges positions; it does not raise numerical values to a power.
An entry at address $(i,j)$ moves to $(j,i)$. Swapping indices twice restores $(i,j)$, proving the double-transpose rule for every entry. A square matrix retains its shape, but off-diagonal entries may exchange positions. Diagonal entries stay put because their indices are equal. Consequently, identical shapes before and after transposition do not establish equality of the matrices themselves. Shape describes available addresses; the entry rule determines which value occupies each address.
Definitions and further reading: Mathematics for Machine Learning.
In plain language
Transposition exchanges the roles of rows and columns while preserving every entry. A row becomes the column with the same index, so the order 2, 4, 6 is not reversed. This is an exact address change rather than a rotation of the paper. In a student-by-subject table, subjects become rows and students become columns; relocation improves no grades.
Example
The example transposes the original illustrative column $(2,4,6)^{\mathsf T}$. The column is followed by the resulting row and the reverse operation.
Transposition: Swapping Rows and Columns
Column
Each value occupies its own row.
Step 1 of 3: Column
Read all steps
Column
Each value occupies its own row.
- x₁
- 2current
- x₂
- 4current
- x₃
- 6current
Row
After exchanging indices, the values share one row.
- xᵀ
- 2current4current6current
Restored column
Transposing again restores the column.
- x₁
- 2current
- x₂
- 4current
- x₃
- 6current
The value four begins in row two, column one, then moves to row one, column two. Two remains the first coordinate and six the third. Transposing back restores each original address. Checking the collection of values alone is insufficient: the row $(6,4,2)$ contains the same numbers but is not the required transpose. Checking both the shape and each swapped address exposes that error.
Related concepts: Matrices.