Suppose that m ×n has full column rank. Prove that the reduced decomposition is unique where m ×n has orthonormal columns and n×n is upper triangular with positive diagonal entries. Do this in steps....

Suppose that

^m
×n
has full column rank. Prove that the reduced decomposition

is unique where

^m
×n
has orthonormal columns and

^n×n
is upper triangular with positive diagonal entries. Do this in steps.

 Show that

<sup>T

T</sup>

Prove that

^T
 is symmetric positive definite, and apply Theorem 13.3 to show that
 is the unique Cholesky factor of

^T

 Show that
 must be unique.

 Give an example to show that there is no guarantee of uniqueness if
 does not have full column rank.