Given a pair ( x , y ) (x, y) (x,y) of unit vectors, define the function normal ( x , y ) \text{normal}(x, y) normal(x,y) to give the unique pair ( x ′ , y ′ ) (x', y') (x′,y′) of unit vectors that satisfy the relations ⟨ x ′ , x ⟩ = 0 , ⟨ y ′ , y ⟩ = 0 , ⟨ x ′ , y ⟩ ≤ 0 , ⟨ y ′ , x ⟩ ≤ 0. Since H H H is ( n − 1 ) (n - 1) (n−1)-dimensional, we have the recurrence F k + 1 n = F k n + F k n − 1 . Then the recurrence above means that, as far as the non-constant terms of F k ( x ) F_k(x) Fk​(x) are concerned, F k + 1 ( x ) = F k ( x ) + x F k ( x ) = ( 1 + x ) F k ( x ) . Given a matrix M M M with dimensions n × 2 n , n \times 2n, n×2n, and let's divide it into blocks M = [ M 0 M 1 ] . Altogether we have argued that the matrix M = [ M 0 M 1 ] M = \begin{bmatrix} M_0 & M_1 \end{bmatrix} M=[M0​​M1​​] contains the origin exactly when M ′ = [ M 0 ⊥ − M 1 ⊥ ] M' = \begin{bmatrix} M_0^\perp & -M_1^\perp \end{bmatrix} M′=[M0⊥​​−M1⊥​​] does not.