Question

Consider an invertible n×nmatrix A. Can you write Aas A=LQ, where Lis a lowertriangular matrix andQis orthogonal? Hint: Consider the QRfactorizationof ${\mathit{A}}^{\mathbf{T}}$.

Short answer

$L=R^T,Q=P^T\mathrm{and}A=LQ$

Step-by-step solution

L is a lower triangular matrix.

Given that A is an n×n invertible matrix. We claimed that there exists lower triangular matrix L and an orthogonal matrix Q such that A=LQ.

Since$detA=det{A}^T$and therefore,$A^T$ is also invertible. Now we have PR factorization for ,i.e.

There exists upper triangular matrix R and an orthogonal matrix P such that

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

By observing in the above factorization. Since R is an upper triangular matrix therefore, will be lower triangular matrix. Also, the matrix $P^T$is orthogonal and hence it has a factorization for A as lower triangular matrix and an orthonormal matrix.

Hence, $L=R^T,Q=P^T\mathrm{and}A=LQ$.