Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Digital systems, linear algebra, introduction to formal proof / 6 / b

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- Solution b

Solution

Use

R_{2} \leftarrow R_{2} + 2 R_{1}

and

R_{3} \leftarrow R_{3} - 3 R_{1}

, followed by

R_{3} \leftarrow R_{3} + R_{2}

. These elementary row operations give

1 - 2 3 4 - 4 8 100 \sim 100 44 - 4 12 - 3 \sim U 100440 12 - 1 .

The elimination multipliers are

m_{21} = - 2

,

m_{31} = 3

,

m_{32} = - 1

. Record them in the unit lower triangular matrix

L

:

A = LU, L = 1 - 2 3 01 - 1 001, U = 100440 12 - 1 .

To see the elementary matrix relation directly, let

F_{21} = I + 2 e_{21}

,

F_{31} = I - 3 e_{31}

and

F_{32} = I + e_{32}

, where

e_{ij}

has a single 1 in position

(i, j)

. Then

U = F_{32} F_{31} F_{21} A

, and reversing and inverting these operations gives

L = F_{21}^{- 1} F_{31}^{- 1} F_{32}^{- 1}

. Direct matrix multiplication also verifies the stated LU decomposition.

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