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Expert-verified Found in: Page 234 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Consider an invertible n ×n matrix A. Can you write A as A = LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of ${{\mathbit{A}}}^{{\mathbf{T}}}$.

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

See the step by step solution

## Step 1: 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

${A}^{T}=PR\phantom{\rule{0ex}{0ex}}={\left({A}^{T}\right)}^{T}={\left(PR\right)}^{T}=A⇒{R}^{T}{P}^{T}\phantom{\rule{0ex}{0ex}}={\left(AB\right)}^{T}={B}^{T}{A}^{T}$

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, $L={R}^{T},Q={P}^{T}\mathrm{and}A=LQ$. ### Want to see more solutions like these? 