Exercise 4.4.25

Answers

This question shows 2 by 2 formulas for $QR$ ; breakdown $R_{22} = 0$ for singular $A$ . Nonsingular example:

[\begin{matrix} 2 & 1 \\ 1 & 1 \end{matrix}] = \frac{1}{\sqrt{5}} [\begin{matrix} 2 & - 1 \\ 1 & 2 \end{matrix}] \cdot \frac{1}{\sqrt{5}} [\begin{matrix} 5 & 3 \\ 0 & 1 \end{matrix}] .

Singular example:

[\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] = \frac{1}{\sqrt{2}} [\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix}] \cdot \frac{1}{\sqrt{2}} [\begin{matrix} 2 & 2 \\ 0 & 0 \end{matrix}] .

The Gram-Schmidt process breaks down when $ad - bc = 0$ .

mathchakra

2023-09-14 08:29