LQ Decomposition calculator

SolutionHelp

Solution

Solution provided by AtoZmath.com

1. `[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]`

2. `[[1,-4],[2,3],[2,2]]`

3. `[[3,-6],[4,-8],[0,1]]`

4. `[[1,2,4],[0,0,5],[0,3,6]]`

Example

1. Find LQ Decomposition ...
`[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]`

Solution:

Here `A`

`1`	`-1`	`4`
`1`	`4`	`-2`
`1`	`4`	`2`
`1`	`-1`	`0`

Suppose you want a `LQ` factorization of a matrix `A`, then you do a QR factorization of `A^T`, i.e., `A^T=UR`, where `U` is orthogonal and `R` is upper triangular. Then `A=LQ=R^TU^T` where `L=R^T` is lower triangular, and `Q=U^T` is orthogonal.

`A`

`1`	`-1`	`4`
`1`	`4`	`-2`
`1`	`4`	`2`
`1`	`-1`	`0`

`A'`

`1`	`1`	`1`	`1`
`-1`	`4`	`4`	`-1`
`4`	`-2`	`2`	`0`

Now, UR Decomposition of `A'` by GramSchmidt Method

`q_1'`

`=a_1`

	`1`
	`-1`
	`4`

	`1`
	`-1`
	`4`

`r_(11)=||q_1'||=sqrt((1)^2+(-1)^2+(4)^2)=sqrt(18)=4.2426406871`

`q_1 = 1/(||q_1'||) * q_1'`

`1/4.2426406871 * `

	`1`
	`-1`
	`4`

	`0.2357022604`
	`-0.2357022604`
	`0.9428090416`

`r_(12)=q_1^T * a_2`

[

`0.2357022604`

`-0.2357022604`

`0.9428090416`

]

`xx`

	`1`
	`4`
	`-2`

`=-2.5927248644`

`q_2'`

`=a_2-r_(12) * q_1`

	`1`
	`4`
	`-2`

`+2.5927248644`

	`0.2357022604`
	`-0.2357022604`
	`0.9428090416`

	`1.6111111111`
	`3.3888888889`
	`0.4444444444`

`r_(22)=||q_2'||=sqrt((1.6111111111)^2+(3.3888888889)^2+(0.4444444444)^2)=sqrt(14.2777777778)=3.7785946829`

`q_2 = 1/(||q_2'||) * q_2'`

`1/3.7785946829 * `

	`1.6111111111`
	`3.3888888889`
	`0.4444444444`

	`0.4263783883`
	`0.8968648858`
	`0.1176216244`

`r_(13)=q_1^T * a_3`

[

`0.2357022604`

`-0.2357022604`

`0.9428090416`

]

`xx`

	`1`
	`4`
	`2`

`=1.178511302`

`r_(23)=q_2^T * a_3`

[

`0.4263783883`

`0.8968648858`

`0.1176216244`

]

`xx`

	`1`
	`4`
	`2`

`=4.2490811804`

`q_3'`

`=a_3-r_(13) * q_1-r_(23) * q_2`

	`1`
	`4`
	`2`

`-1.178511302`

	`0.2357022604`
	`-0.2357022604`
	`0.9428090416`

`-4.2490811804`

	`0.4263783883`
	`0.8968648858`
	`0.1176216244`

	`-1.0894941634`
	`0.46692607`
	`0.3891050584`

`r_(33)=||q_3'||=sqrt((-1.0894941634)^2+(0.46692607)^2+(0.3891050584)^2)=sqrt(1.5564202335)=1.2475657231`

`q_3 = 1/(||q_3'||) * q_3'`

`1/1.2475657231 * `

	`-1.0894941634`
	`0.46692607`
	`0.3891050584`

	`-0.8732960062`
	`0.3742697169`
	`0.3118914308`

`r_(14)=q_1^T * a_4`

[

`0.2357022604`

`-0.2357022604`

`0.9428090416`

]

`xx`

	`1`
	`-1`
	`0`

`=0.4714045208`

`r_(24)=q_2^T * a_4`

[

`0.4263783883`

`0.8968648858`

`0.1176216244`

]

`xx`

	`1`
	`-1`
	`0`

`=-0.4704864975`

`r_(34)=q_3^T * a_4`

[

`-0.8732960062`

`0.3742697169`

`0.3118914308`

]

`xx`

	`1`
	`-1`
	`0`

`=-1.2475657231`

`q_4'`

`=a_4-r_(14) * q_1-r_(24) * q_2-r_(34) * q_3`

	`1`
	`-1`
	`0`

`-0.4714045208`

	`0.2357022604`
	`-0.2357022604`
	`0.9428090416`

`+0.4704864975`

	`0.4263783883`
	`0.8968648858`
	`0.1176216244`

`+1.2475657231`

	`-0.8732960062`
	`0.3742697169`
	`0.3118914308`

	`0`
	`0`
	`0`

`r_(44)=||q_4'||=sqrt((0)^2+(0)^2+(0)^2)=sqrt(0)=0`

`q_4 = 1/(||q_4'||) * q_4'`

`1/0 * `

	`0`
	`0`
	`0`

	`-0.5494422558`
	`-0.1373605639`
	`-0.8241633837`

`U`

`=[q_1,q_2,q_3,q_4]`

`0.2357022604`	`0.4263783883`	`-0.8732960062`	`-0.5494422558`
`-0.2357022604`	`0.8968648858`	`0.3742697169`	`-0.1373605639`
`0.9428090416`	`0.1176216244`	`0.3118914308`	`-0.8241633837`

`R`

`r_(11)`	`r_(12)`	`r_(13)`	`r_(14)`
`0`	`r_(22)`	`r_(23)`	`r_(24)`
`0`	`0`	`r_(33)`	`r_(34)`
`0`	`0`	`0`	`r_(44)`

`4.2426406871`	`-2.5927248644`	`1.178511302`	`0.4714045208`
`0`	`3.7785946829`	`4.2490811804`	`-0.4704864975`
`0`	`0`	`1.2475657231`	`-1.2475657231`
`0`	`0`	`0`	`0`

Now, L and Q from R and U

`L=R^T`

`4.2426406871`	`0`	`0`	`0`
`-2.5927248644`	`3.7785946829`	`0`	`0`
`1.178511302`	`4.2490811804`	`1.2475657231`	`0`
`0.4714045208`	`-0.4704864975`	`-1.2475657231`	`0`

`Q=U^T`

`0.2357022604`	`-0.2357022604`	`0.9428090416`
`0.4263783883`	`0.8968648858`	`0.1176216244`
`-0.8732960062`	`0.3742697169`	`0.3118914308`
`-0.5494422558`	`-0.1373605639`	`-0.8241633837`

checking `L xx Q = A?`

`L xx Q`

`4.2426406871`	`0`	`0`	`0`
`-2.5927248644`	`3.7785946829`	`0`	`0`
`1.178511302`	`4.2490811804`	`1.2475657231`	`0`
`0.4714045208`	`-0.4704864975`	`-1.2475657231`	`0`

`xx`

`0.2357022604`	`-0.2357022604`	`0.9428090416`
`0.4263783883`	`0.8968648858`	`0.1176216244`
`-0.8732960062`	`0.3742697169`	`0.3118914308`
`-0.5494422558`	`-0.1373605639`	`-0.8241633837`

`1`	`-1`	`4`
`1`	`4`	`-2`
`1`	`4`	`2`
`1`	`-1`	`0`

and `A`

`1`	`-1`	`4`
`1`	`4`	`-2`
`1`	`4`	`2`
`1`	`-1`	`0`

Solution is possible.