[Math] General Form of Orthogonal Upper Triangular Matrices

linear algebramatricesorthogonality

I have the following 2-part question:

  1. Find all $n \times n$ matrices that are both orthogonal and upper triangular, with positive diagonal entries.

  2. Show that the $QR$ factorization of an invertible $n \times n$ matrix is unique. Hint: if $A=Q_1R_1=Q_2R_2$, then the matrix $Q_2^{-1}Q_1=R_2R_1^{-1}$ is both orthogonal and upper triangular, with positive diagonal entries.

I realize that the general form of the $R$ matrix is upper triangular, with diagonal entries as vector lengths, which by definition must be positive. I'm not sure about the big picture though. Anyone kind enough to nudge me in the right direction? Thanks!

Best Answer

(1) There's nothing simpler than to realize that $Q^T=Q^{-1}$ implies that $Q$ is both upper and lower triangular. Hence diagonal.

(2) Note that $Q_2^TQ_1=R_2R_1^{-1}$ implies that $R_2R_1^{-1}$ is

  1. orthogonal -- because it is a product of two orthogonal matrices,
  2. diagonal -- because it is orthogonal and triangular.

The only real orthogonal diagonal matrix is a matrix with $\pm 1$ on the diagonal. Consequently, for any two QR factorizations of $A$, the R-factors are related by such a simple diagonal matrix and in particular, their diagonals differ just by the sign. Once you require the positive entries on the diagonal of $R$, it hence follows that $R_1=R_2$ since the diagonal of $R_2R_1^{-1}$ is nothing but the "componentwise ratio" of the diagonals of $R_1$ and $R_2$.

Related Question