38. Transpose

The transpose ${𝐴}^{𝑇}$ of a matrix $𝐴$ flips it across its main diagonal — rows become columns:

If $𝐴$ is $𝑚\times 𝑛$, then ${𝐴}^{𝑇}$ is $𝑛\times 𝑚$.

38.1. Properties

• Involution: ${\left({𝐴}^{𝑇}\right)}^{𝑇}=𝐴$
• Sum: ${\left(𝐴+𝐵\right)}^{𝑇}={𝐴}^{𝑇}+{𝐵}^{𝑇}$
• Scalar: ${\left(𝑐𝐴\right)}^{𝑇}=𝑐{𝐴}^{𝑇}$
• Product (order reversed): ${\left(𝐴𝐵\right)}^{𝑇}={𝐵}^{𝑇}{𝐴}^{𝑇}$
• Inverse: ${\left({𝐴}^{𝑇}\right)}^{-1}={\left({𝐴}^{-1}\right)}^{𝑇}$ (when $𝐴$ is invertible)
• Determinant: $det\left({𝐴}^{𝑇}\right)=det\left(𝐴\right)$
• Rank: $\text{rank}\left({𝐴}^{𝑇}\right)=\text{rank}\left(𝐴\right)$

38.2. Special cases

• Symmetric: ${𝐴}^{𝑇}=𝐴$
• Skew-symmetric: ${𝐴}^{𝑇}=-𝐴$
• Orthogonal: ${𝐴}^{𝑇}={𝐴}^{-1}$, i.e. ${𝐴}^{𝑇}𝐴=𝐼$
• Vector transpose: a column vector becomes a row vector and vice-versa

38.3. Connection to dot product

For column vectors $\overrightarrow{𝑢},\overrightarrow{𝑣}\in {ℝ}^{𝑛}$:

(see Dot Product)