Question

Consider a line Lin ${\mathit{R}}^{\mathbf{n}}$, spanned by a unit vector

$\overrightarrow{\mathbf{u}}\mathbf{=}\left[\begin{array}{c}{u}_1\\ u_2\\ .\\ .\\ .\\ u_n\end{array}\right]$

Consider the matrix Aof the orthogonal projection onto L. Describe the ijth entry of A, in terms of the components${\overrightarrow{\mathbf{u}}}_{\mathbf{i}}$of $\overrightarrow{\mathbf{u}}$.

Short answer

The ijth entry of A is given by ${\overrightarrow{u}}_1.{\overrightarrow{u}}_j$.

Step-by-step solution

The matrix of an orthogonal projection.

The matrix P of the orthogonal projection onto V=spanis $\left\{{\overrightarrow{u}}_1{\overrightarrow{u}}_2,.....{\overrightarrow{u}}_m\right\}$given by, $\mathit{P}\mathbf{=}\mathit{Q}{\mathit{Q}}^{\mathbf{T}}$where$\mathit{Q}\mathbf{=}\mathbf{\{}{\overrightarrow{\mathbf{u}}}_{\mathbf{1}}\mathbf{}{\overrightarrow{\mathbf{u}}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}{\overrightarrow{\mathbf{u}}}_{\mathbf{m}}\mathbf{\}}$.

Where the spanning set is orthonormal basis.

Thus, in our case the vector subspace is L and therefore, the required projection matrix will be,

$$\begin{gathered} A=Q{Q}^T \\ =\left[\begin{array}{c}{\mathrm{u}}_1\\ {\mathrm{u}}_2\\ .\\ .\\ .\\ {\mathrm{u}}_{\mathrm{n}}\end{array}\right]\left[\begin{array}{cc}{u}_1& u_2.....u_m\end{array}\right] \\ =\left[\begin{array}{cc}{u}_1^2& u_2{u}_1......u_1{u}_n\\ u_2{u}_1& u_1^2.........u_2{u}_n\\ .& .........\\ .& .........\\ .& .......\\ u_n{u}_1& u_2{u}_n.......u_n{u}_n\end{array}\right] \end{gathered}$$

Hence, the ijth entry of the matrix A is given by ${\overrightarrow{u}}_i{\overrightarrow{u}}_j$.