(a) Compute P = QQ^T when q1 = (.8, .6,0) and q2 = (-.6, .8,0). Verify that P^2 = P. (b) Prove that always (QQ^T)^2 = QQ^T by using Q^T Q = I. Then P = QQ^T is the projection matrix onto the column space of Q.

Question

(a) Compute P = Q{Q}^{T} when {q}_{1} = (.8, .6,0) and {q}_{2} = (-.6, .8,0). Verify that {P}^{2} = P. (b) Prove that always (Q{Q}^{T})^{2} = Q{Q}^{T} by using {Q}^{T}Q = I. Then P = Q{Q}^{T} is the projection matrix onto the column space of Q.

Accepted Answer

(a) [latex]Q =\left[ \begin{matrix} .8 & -.6 \ .6 & .8 \ 0 & 0 \end{matrix} ight][/latex] has [latex]P = Q{Q}^{T}=\left[ \begin{matrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 0 \end{matrix} ight] [/latex]

(b) [latex](Q{Q}^{T})(Q{Q}^{T}) = Q({Q}^{T}Q){Q}^{T} ={Q}^{T}Q[/latex].

Question : (a) Compute P = QQ^T when q1 = (.8, .6,0) and q2 = (-.6, .8,...