Alex Loftus
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Transposition changes how information is grouped

The x-values (a, c, e) form one 3D vector; the y-values (b, d, f) form the other. The original vector order becomes the new coordinate order.
The x-values (a, c, e) form one 3D vector; the y-values (b, d, f) form the other. The original vector order becomes the new coordinate order.

Regroup the same numbers

Start with three 2D vectors written as rows:

A=[123456].A=\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}.

Each row can be drawn as a point in a plane: (1,2)(1,2), (3,4)(3,4), and (5,6)(5,6). Now collect the first coordinate of every point into one row, and the second coordinate into another:

AT=[135246].A^\mathsf{T}=\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}.

The new rows can be drawn as two points in 3D. These are two geometric interpretations of the same six numbers.

Follow one entry

The 4 began in row 2, column 2, and stays there. The 5 moves from row 3, column 1 to row 1, column 3. In general,

(AT)ij=Aji.(A^\mathsf{T})_{ij}=A_{ji}.

Transposing twice puts every entry back: (AT)T=A(A^\mathsf{T})^\mathsf{T}=A.

Why the new picture matters

Suppose rows describe observations and columns describe features. Looking across a row shows one observation. Looking down a column shows how one feature varies across observations. Transposition switches which grouping is convenient to read or multiply.

Every entry remains available. This differs from a projection, which can discard coordinates and merge distinct inputs into one output.

Track a shape

What shape does a 100×4100\times4 matrix have after transposition?

Answer: 4×1004\times100. It still contains 400 entries, with row and column roles exchanged.