Question: Given u in R n with u T u = 1, let P = uu T (an outer product) and Q = I - 2P.

Given u in Rⁿ with u^Tu = 1, let P = uu^T(an outer product) and Q = I - 2P. Justify statements,

(a) P²= P

(b) P^T = P

The transformation x ↦ Px is called a projection, and x ↦ Qx is called a Householder reflection. Such reflections are used in computer programs to create multiple zeros in a vector (usually a column of a matrix).

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