5.110 min read

The Transpose

The transpose of a matrix flips it across its main diagonal. Every row becomes a column and every column becomes a row. If $A$ is an $m \times n$ matrix, then $A^t$ is an $n \times m$ matrix, and the entry in row $i$ , column $j$ of $A^t$ equals the entry in row $j$ , column $i$ of $A$ .

A column vector $\mathbf{v} \in \mathbb{R}^m$ is just an $m \times 1$ matrix. Its transpose $\mathbf{v}^t$ is a $1 \times m$ matrix — we call this a row vector. Row vectors let us write dot products as matrix products: $\mathbf{u}^t \mathbf{v} = \mathbf{u} \cdot \mathbf{v}$ .

Up to this point, we have studied matrices entirely through their columns. The transpose lets us apply everything we know about column structure to the rows of the original matrix — it is our gateway to studying row space.

Formal View

Definition 5.1 — Transpose

Let

A

be an

m \times n

matrix. Its transpose

A^t

is the

n \times m

matrix defined by

(A^t)_{ij} = A_{ji}

. For a column vector

\mathbf{v} \in \mathbb{R}^m

, the transpose

\mathbf{v}^t

is a

1 \times m

row vector.

Remark 5.1

Transposing twice returns the original:

(A^t)^t = A

. The transpose of a product reverses the order:

(AB)^t = B^t A^t

. You can verify this by checking dimensions: if

A

m \times n

and

B

n \times p

, then

(AB)^t

p \times m = B^t A^t

Interactive Visualization

Transpose Visualizer

Why This Matters

The transpose is fundamental to nearly every branch of applied mathematics.

Symmetric matrices $A = A^t$ arise in physics, statistics, and optimization — covariance matrices and moment-of-inertia tensors are all symmetric
In machine learning, the normal equations for least-squares regression involve $A^t A$ and $A^t \mathbf{b}$
Quantum mechanics uses Hermitian matrices (the complex analog of symmetric), whose transpose plays a central role
The transpose determines the "adjoint" of a linear map, connecting input and output spaces in a dual relationship

Learning Resources

Transpose of a matrix

Khan Academy

Clear introduction to the mechanics of transposing a matrix.

5 min

Lecture 14: Orthogonal vectors and subspaces

MIT OpenCourseWare

Gilbert Strang introduces the transpose in the context of orthogonality.

47 min

Quiz

Question 1

If $A$ is a $3 \times 5$ matrix, what are the dimensions of $A^t$ ?

Question 2

$(AB)^t = A^t B^t$ for all matrices $A$ and $B$ with compatible dimensions.

Question 3

Which expression equals the dot product $\mathbf{u} \cdot \mathbf{v}$ using matrix notation?

Common Mistakes

Writing $(AB)^t = A^t B^t$ instead of $B^t A^t$ — the transpose reverses order, just as the inverse does.
Confusing a row vector $\mathbf{v}^t$ with a column vector — they have different dimensions and behave differently in matrix products.
Assuming $(A^t)^{-1} = (A^{-1})^t$ without knowing $A$ is invertible — this identity only holds for square invertible matrices.