The Transpose
The transpose of a matrix flips it across its main diagonal: rows become
columns and columns become rows. We mark it with a small \mathsf{T}:
A = \begin{bmatrix} 2 & -1 & 0 \\ 3 & 5 & 4 \end{bmatrix} \;\longrightarrow\; A^{\mathsf{T}} = \begin{bmatrix} 2 & 3 \\ -1 & 5 \\ 0 & 4 \end{bmatrix}.
Entry by entry, the rule is simply (A^{\mathsf{T}})_{ij} = A_{ji} —
swap the row and column index. A 2\times 3 matrix transposes into a
3\times 2 one; the shape turns on its side.
Watch the flip
Step through the entries. Each highlighted number in A lands in the
mirror-image position of A^{\mathsf{T}} — row and column swapped. The
diagonal entries stay put; everything else trades places across it.
Why it earns its keep
The transpose is the bridge between rows and columns, so it turns up wherever both appear at
once. The dot product
of two column vectors is secretly \vec{u}^{\mathsf{T}}\vec{v} — a row
times a column. And a matrix equal to its own transpose
(A = A^{\mathsf{T}}) is called
symmetric,
a beautifully well-behaved type that powers
PCA.