Transpose

A transpose exchanges rows with columns in a matrix such that the entry $A\_ij=A^T\_ji$.

• $(AB{)}^T=B^T{A}^T$
• $(A+B{)}^T=A^T+B^T$
• $(A^{-1}{)}^T=(A^T{)}^{-1}$ A transpose is the unique matrix satisfying $$(Ax)\cdot y=x\cdot (A^{T}y)$$

Definition of Outer and Inner Products

Suppose $x=\left[\begin{array}{c}x\_1x\_2\end{array}\right]$ and $y=\left[\begin{array}{c}y\_1y\_2\end{array}\right]$ are being multiplied.

• The dot-product of the vectors $x\cdot y$ is $x^{T}y$ because $\text{row}\times \text{col}=\text{number}$
• The inner-product $x{y}^T$ produces a matrix