Numerical Analysis | Part IB, 2001

(a) Let BB be an n×nn \times n positive-definite, symmetric matrix. Define the Cholesky factorization of BB and prove that it is unique.

(b) Let AA be an m×nm \times n matrix, mnm \geqslant n, such that rankA=n\operatorname{rank} A=n. Prove the uniqueness of the "skinny QR factorization"


where the matrix QQ is m×nm \times n with orthonormal columns, while RR is an n×nn \times n upper-triangular matrix with positive diagonal elements.

[Hint: Show that you may choose RR as a matrix that features in the Cholesky factorization of B=ATAB=A^{T} A.]

Typos? Please submit corrections to this page on GitHub.