Transpose of a Matrix: Notes and Theorems

These are introductory notes on the transpose of a matrix. They cover its definition, the order of the transpose, its main properties, symmetric and skew symmetric matrices, and the proofs of two theorems. Worked problems are left for the next video.

What the transpose is

The transpose of a matrix is obtained by interchanging its rows and columns. For a matrix $A$, the transpose is written $A^{\mathsf T}$ (also denoted $A'$).

If $A$ is an $m \times n$ matrix, then swapping the rows and columns gives an $n \times m$ matrix, so the order of $A^{\mathsf T}$ is $n \times m$.

$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \quad (2 \times 3) \implies A^{\mathsf T} = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix} \quad (3 \times 2)$

Each row of $A$ becomes a column of $A^{\mathsf T}$.

Properties of the transpose

For matrices of suitable order and a scalar $k$:

$\left(A^{\mathsf T}\right)^{\mathsf T} = A$

$\left(kA\right)^{\mathsf T} = k\,A^{\mathsf T}$

$\left(A + B\right)^{\mathsf T} = A^{\mathsf T} + B^{\mathsf T} \quad (A \text{ and } B \text{ of the same order})$

$\left(AB\right)^{\mathsf T} = B^{\mathsf T} A^{\mathsf T}$

Transposing twice returns the original matrix, and the transpose of a product reverses the order of the factors.

Symmetric and skew symmetric matrices

These definitions apply only to square matrices, where the number of rows equals the number of columns.

• A square matrix $A$ is symmetric if $A^{\mathsf T} = A$.
• A square matrix $A$ is skew symmetric if $A^{\mathsf T} = -A$.

A useful fact: in a skew symmetric matrix every diagonal element is zero. This follows because each diagonal entry must equal its own negative.

Theorem 1 and its proof

For any square matrix $A$, the matrix $A + A^{\mathsf T}$ is symmetric and the matrix $A - A^{\mathsf T}$ is skew symmetric.

Sum is symmetric

Let $B = A + A^{\mathsf T}$. We show $B^{\mathsf T} = B$.

$B^{\mathsf T} = \left(A + A^{\mathsf T}\right)^{\mathsf T} = A^{\mathsf T} + \left(A^{\mathsf T}\right)^{\mathsf T}$

Using $\left(A^{\mathsf T}\right)^{\mathsf T} = A$:

$B^{\mathsf T} = A^{\mathsf T} + A = A + A^{\mathsf T} = B$

Since $B^{\mathsf T} = B$, the matrix $A + A^{\mathsf T}$ is symmetric.

Difference is skew symmetric

Let $C = A - A^{\mathsf T}$. We show $C^{\mathsf T} = -C$.

$C^{\mathsf T} = \left(A - A^{\mathsf T}\right)^{\mathsf T} = A^{\mathsf T} - \left(A^{\mathsf T}\right)^{\mathsf T} = A^{\mathsf T} - A$

Taking $-1$ out as a common factor:

$C^{\mathsf T} = -\left(A - A^{\mathsf T}\right) = -C$

Since $C^{\mathsf T} = -C$, the matrix $A - A^{\mathsf T}$ is skew symmetric.

Theorem 2: splitting any square matrix

Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix:

$A = \tfrac{1}{2}\left(A + A^{\mathsf T}\right) + \tfrac{1}{2}\left(A - A^{\mathsf T}\right)$

Expanding the right side gives $\tfrac{1}{2}A + \tfrac{1}{2}A^{\mathsf T} + \tfrac{1}{2}A - \tfrac{1}{2}A^{\mathsf T} = A$, so the identity holds. By Theorem 1, $\tfrac{1}{2}\left(A + A^{\mathsf T}\right)$ is symmetric and $\tfrac{1}{2}\left(A - A^{\mathsf T}\right)$ is skew symmetric.

One further result worth remembering links the transpose and the inverse:

$\left(A^{\mathsf T}\right)^{-1} = \left(A^{-1}\right)^{\mathsf T}$

Key takeaways

• The transpose swaps rows and columns; an $m \times n$ matrix becomes $n \times m$, and transposing twice returns the original.
• A square matrix is symmetric when $A^{\mathsf T} = A$ and skew symmetric when $A^{\mathsf T} = -A$; every skew symmetric matrix has zeros on its diagonal.
• For any square matrix, $A + A^{\mathsf T}$ is symmetric and $A - A^{\mathsf T}$ is skew symmetric, and $A$ can be split as the sum of these two halves.