Linear Algebra and Its Applications If u is a unit vector sh

(Linear Algebra and Its Applications) If u is a unit vector, show that Q = I - 2uu^T is a symmetric orthogonal matrix.

Solution

10. Let u be a unit vector in R ⁿ

and let Q = I 2 u u^T .

Proof. Since Q^T = (I 2 u u^T ) ^T

= I ^T 2(u u^T ) ^T

= I 2(u ^T ) ^T u ^T

= I 2 u uT = Q, by definition,

therfore Q is symmetric.

As u is a unit vector hence u ^T u = [[ u ]] ² = 1.

So Q^T Q = QQ = (I 2 u u^T )(I 2 u u^T )

= I 2 u u^T 2 u u^T + 4(u u^T )(u u^T )

= I 4 u uT + 4 u(u T u)u T

= I 4 u uT + 4 u uT

= I.

therfore by this Q is an orthogonal matrix.