Systems and matrices · Definition

Transpose matrix

linear algebramatrices

Before you start you need to know

The transpose swaps rows and columns; it is a simple operation, but it singles out symmetric matrices and appears in almost every formula of the theory.

Definition

The transpose \(A^T\) of a matrix \(A\) is the one obtained by exchanging its rows and columns.

\[(A^T)_{ij}=a_{ji}\]

Transposing a matrix means reflecting it across its main diagonal: row \(i\) becomes column \(i\). For vectors, a column vector becomes a row vector, which is where the dot product written as \(\mathbf{u}^T\mathbf{v}\) comes from. The transpose is therefore the bridge between «rows» and «columns», two ways of reading the same matrix.

The least intuitive property is \((AB)^T=B^TA^T\): the order reverses. Sizes explain it: if \(A\) is \(m\times n\) and \(B\) is \(n\times p\), then \(B^T\) is \(p\times n\) and \(A^T\) is \(n\times m\), so only the product \(B^TA^T\) is defined. It is the same order reversal that happens with inverses, and that is no coincidence.

Guiding example: if \(A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}\), then \(A^T\) is the \(3\times2\) matrix with rows \((1,4)\), \((2,5)\), \((3,6)\). A matrix is symmetric if \(A^T=A\); these arise when studying quadratic forms and eigenvalues. Typical mistake: believing \((AB)^T=A^TB^T\), which is almost never true (it does not even make sense with different sizes).

Think about it

If \(A\) is \(3\times 5\), what size is \(A^TA\)?

\(5\times5\): \(A^T\) is \(5\times3\) and \(A\) is \(3\times5\); moreover \(A^TA\) is always symmetric.

What happens to the main diagonal of a matrix when it is transposed?

It does not change: the entries \(a_{ii}\) stay in place, because \((A^T)_{ii}=a_{ii}\).

Exercises 10

  1. Compute the transpose of \(\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{pmatrix}\).
  2. If \(A\) is a \(2\times4\) matrix, what size is \(A^T\)?
  3. Which of these matrices is symmetric (satisfies \(A=A^T\))?
  4. If \(A^T=\begin{pmatrix}1 & 3 \\ 2 & 4\end{pmatrix}\), what is \(A\)?
  5. If \(A\) and \(B\) have the same size, what is \((A+B)^T\) equal to?
  6. With \(A=\begin{pmatrix}1 & 2 \\ 0 & 1\end{pmatrix}\) and \(B=\begin{pmatrix}3 & 0 \\ 1 & 2\end{pmatrix}\), compute \((AB)^T\).
  7. Compute \(A+A^T\) for \(A=\begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix}\).
  8. Let \(\mathbf{x}=(1,-2,3)^T\) be a column vector. What is \(\mathbf{x}^T\mathbf{x}\)?
  9. A square matrix satisfies \(A^T=-A\) (skew-symmetric). What can be said about its main diagonal?
  10. Let \(A\) be an \(m\times n\) matrix. What can be said about \(A^TA\)?

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §2.7
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §3.E