Definition

Matrix transpose is found by interchanging the rows and columns of the matrix.

• In other words, the transpose of $m \times n$ $A$ matrix = $n \times m$ $B$ matrix.
• It can denote the transpose of matrix $A$ as $A^T$

Example

• Here is an 2x2 $A$ matrix.

  $$ A =\begin{bmatrix} a & b & c \\ d & e & f \\

  \end{bmatrix} $$

• $A^T$ is the transpose of the $A$ matrix.

$$ A^T =\begin{bmatrix} a & d \\ b & e \\ c & f \end{bmatrix} $$

Properties

Transopose of Transpose

• When you transpose a matrix twice, you'll get back to the original matrix.

  $$ (A^T)^T = A $$

• This means that the transpose operation is self-inverse

Other Properties

You can observe other applied properties of the transpose.

• $(cA)^T = cA^T$
• $(A + B)^T = A^T + B^T$
• $(AB)^T = B^TA^T$
• $(A^{-1})^T = (A^T)^{-1}$
• if $A \cdot B = 0$ , $A \cdot B = A^T B$

Practice with Python

# 2차원 배열 생성
A = np.array([[1, 2, 3],
              [4, 5, 6]])

# 트랜스포즈
transposed_A = np.transpose(A)

print(transposed_A)

You can watch the result!

**[[1 4]
 [2 5]
 [3 6]]**