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Q48E

Expert-verifiedFound in: Page 234

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Consider an invertible n **

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

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

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

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

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