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Exercise 2.7.26

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(a)

The matrix

E_{21} = [\begin{matrix} 1 \\ - 3 & 1 \\ 1 \end{matrix}]

puts 0 in the 2,1 entry of $E_{21} A$ . Then,

E_{21} A E_{21}^{T} = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 4 \\ 0 & 4 & 9 \end{matrix}]

is still symmetric, with zero also in its 1,2 entry.

(b)

Now use

E_{32} = [\begin{matrix} 1 \\ 1 \\ - 2 & 1 \end{matrix}]

to make the 3,2 entry zero and $E_{32} E_{21} A E_{21}^{T} E_{32}^{T} = D$ also has zero in its 2,3 entry. Key point: Elimination from both sides (rows + columns) gives the symmetric $LD L^{T}$ .

mathchakra

2022-01-16 12:39

Exercise 2.7.26

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