Question

Consider an invertible n×nmatrix A. Can you write A=RQ, where Ris an upper triangular matrix and Q is orthogonal?

Short answer

$R=L^T,Q=P^T$and A = RQ

Step-by-step solution

R is an upper triangular matrix.

It claimed that there exists upper triangular matrix R and an orthogonal matrix Q such that A= RQ.

Recall that, any n× nmatrix A with rank (A)=m, it is always possible to write A=QL where Q is an n× northogonal matrix and L is an n× nlower triangular matrix with positive diagonals.

Since, A is an invertible and so is $A^T$and the rank of A is n.

It will find an orthogonal matrix P and a lower triangular matrix L,

$$\begin{gathered} A^T=PL \\ \Rightarrow \left(A^T\right)={\left(PL\right)}^T \\ \Rightarrow A=L^T{P}^T{\left(AB\right)}^T=B^T{A}^T \end{gathered}$$

By observing in the above factorization. Since L is a lower triangular matrix therefore, $L^T$will be upper triangular matrix. Also, the matrix $P^T$ is orthogonal and hence we have a factorization for A.

Hence, $R=L^T,Q=P^T$and A = RQ is proved.