Question

Assume P is an orthogonal matrix prove that - ⟨Pu,Pv⟩=⟨u,v⟩, for all u,v∈V. - ∥Pu∥=∥u∥ for all u∈V.

Answer 1

An orthogonal matrix P satisfies ⟨Pu,Pv⟩ = ⟨u,v⟩ and ||Pu|| = ||u|| for all u and v in V.

Step-by-step explanation:

An orthogonal matrix P is a square matrix whose columns and rows are orthogonal unit vectors. To prove the given statements:

1. ⟨Pu,Pv⟩ = ⟨u,v⟩:

Proof:

Since P is an orthogonal matrix, its transpose, P^T, is also orthogonal. So,
`- ⟨Pu,Pv⟩ = - (Pu)^T(Pv) = - u^T(P^T)(Pv) = -`
`u^T(P^TP)v = - u^TIv = - ⟨u,v⟩ = ⟨u,v⟩`.
`||Pu|| = ||u||:`

Proof:

Since P is an orthogonal matrix,
`P^T = P^(-1).`
`Pu|| = sqrt((Pu)^TPu) = sqrt(u^T(P^T)(Pu)) = sqrt(u^TIu) = sqrt(⟨u,u⟩) = ||u||`