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Exercise 4.4.19

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If $A = QR$ then $A^{T} A = R^{T} Q^{T} QR = R^{T} R =$ lower triangular times upper triangular (this Cholesky factorization of $A^{T} A$ uses the same $R$ as Gram-Schmidt!). The example has

A = [\begin{matrix} - 1 & 1 \\ 2 & 1 \\ 2 & 4 \end{matrix}] = \frac{1}{3} [\begin{matrix} - 1 & 2 \\ 2 & - 1 \\ 2 & 2 \end{matrix}] [\begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}] = QR

and the same $R$ appears in

A^{T} A = [\begin{matrix} 9 & 9 \\ 9 & 18 \end{matrix}] = [\begin{matrix} 3 & 0 \\ 3 & 3 \end{matrix}] [\begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}] = R^{T} R .

mathchakra

2023-09-14 08:27

Exercise 4.4.19

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