Showing an orthogonalisation process

Question

Can anyone show that:

$\mathbf{a_{\perp}}=\mathbf{a}-\frac{\mathbf{x}\mathbf{x}^T}{\mathbf{x}^T\mathbf{x}}\mathbf{a}$,

$\mathbf{a}\in\mathbb{R}^N$, $\mathbf{x}=(1,1,\dots,1)^T\in\mathbb{R}^N$

results in $\mathbf{a}$ becoming orthogonal to $\mathbf{x}$?

Thank you for your help.

Answer 1

We have that

$$\left(\mathbf{a}-\frac{\mathbf{x}\mathbf{x}^T}{\mathbf{x}^T\mathbf{x}}\mathbf{a}\right)\cdot \mathbf{x}=\mathbf{a}^T\mathbf{x}-\mathbf{a}^T\frac{\mathbf{x}\mathbf{x}^T}{\mathbf{x}^T\mathbf{x}}\mathbf{x}=\mathbf{a}^T\mathbf{x}-\mathbf{a}^T\mathbf{x}=0$$

Refer also to the related

• Writing projection in terms of projection matrix
• orthogonal projection from one vector onto another