> [!summary] The transpose of a matrix is changing the rows to columns > **Key Equations:** > Transposing a matrix in general: > $(A^T) _{ij} = (A)_{ji}$ > Properties of Transposing Matrix: > Double transpose - $(A^T)^T=A$ Sum of transpose - $(A+B)^T=A^T+B^T$ Scalar multiple transpose - $(sA)^T = sA^T$ Product of transpose - $(AB)^T = A^T B^T$ >[!info]+ Read Time **⏱ 1 min** # Definition Transposing a matrix is changing a [[Matrix Notation|matrix]] with $n$ rows to $n$ columns for any matrix in $\mathbb{R}^n$. This transposition is reversible; a matrix can also be changed from $n$ columns to $n$ rows. The [[Dimensions of a Matrix|dimensions]] of a matrix get flipped. If a matrix is denoted as $A \in \mathbb{R}^n$, then the transpose of that matrix is $A^T\in \mathbb{R}^m$ formally defined below. **General $m \times n$ matrix:** $ A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \\ a_{21} & a_{22} & \dots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}, \quad A \in \mathbb{R}^n $ **Transpose $n \times m$ matrix:** $ A^T = \begin{bmatrix} a_{11} & a_{21} & \dots & a_{m1} \\ a_{12} & a_{22} & \dots & a_{m2} \\ \vdots & \vdots & \ddots & \vdots \\ a_{1n} & a_{2n} & \dots & a_{nm} \end{bmatrix}, \quad A^T\in \mathbb{R}^m $ More generally, transposing a matrix is said as using [[Matrix Notation|entries]] in a matrix stated as $ (A^T) _{ij} = (A)_{ji} $ # Properties of Transposing Matrices > [!warning] Assumptions Some important properties are stated below. Each property is proven in [[!! Proof of Properties of Transposing Matrices|proof of properties of transposing matrices]], and are only explained and given here. Keep note of the following assumptions: > - $A$ and $B$ are [[Matrix Notation|matrices]] > - $s$ are any scalar real number ($s\in \mathbb{R}$ The **double transpose** is when a matrix is double transposed, like $(A^T)^T$, the result is the original matrix $A$. Transposing a **sum of matrices** like $(A+B)^T$ results in distributing the transpose over addition. The result is $A^T + B^T$. Similarly **transposing a scalar multiple and a matrix together** results in factoring out the scalar to get the result $(sA)^T=sA^T$ **Transposing a product of matrices** like $(AB)^T$ results in distributing the transpose to both matrices. The result is as follows $(AB)^T=A^TB^T$