Lecture 2

HOSVD bases (Hackbusch)
1) Assume r D = 1. Then
C (D,1) = Σα := diag{σ (D)
1 , . . .},
where σ (D)
i are the singular values of Mα1 (v)
2) For non-leaf vertices α ∈ TD, α �= D, we have
�rα ℓ=1 (σ(α)
ℓ ) 2 C (α,ℓ) C (α,ℓ)H = Σ2 α1 ,
�rα ℓ=1 (σ(α)
ℓ ) 2 C (α,ℓ)TC (α,ℓ) = Σ2 α2 ,
where α1, α2 are the first and second son of α ∈ TD and Σα the i
diagonal of the singular values of Mα (v).
i