orthogonal decomposition and projection

Orthogonal decomposition

The parallel component

In orthogonal decomposition of a target vector $t$ the parallel component $t_{\parallel r}$ is represented by a fraction $\beta$ of the reference vector $r$. $\beta r$ is the vector along $r$ where the difference with $t$, the vector $t-\beta r$, forms a right angle, or in other words the dot product between the two is zero.

$$r\cdot (t-\beta r)=0$$ $$r\cdot t-r\cdot \beta r=0$$ $$\beta =\frac{r\cdot t}{r\cdot r}$$ $$\beta r=\frac{r\cdot t}{r\cdot r}r$$

where $\beta r$ is the parallel component $t_{\parallel r}$.

The perpendicular component

The perpendicular component is found by considering the orthogonal decomposition identity

$$t=t_{\perp r}+t_{\parallel r}$$ $$t_{\perp r}=t-t_{\parallel r}$$ $$x=\underbrace{x_{\text{row}}}\text{projection onto row space}+\underset{\text{perpendicular (lives in null space)}}{\underbrace{x\text{null}}}$$