Find QR Decomposition (Gram Schmidt Method) ...
`[[3,2,4],[2,0,2],[4,2,3]]`

Solution:

Here `A`

=

`3` `2` `4` `2` `0` `2` `4` `2` `3`

`q_1'`

`=a_1`

=

`3` `2` `4`

=

`3` `2` `4`

`r_(11)=||q_1'||=sqrt(3^2+2^2+4^2)=sqrt(29)=5.3852`

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

=

`1/5.3852 * `

`3` `2` `4`

=

`0.5571` `0.3714` `0.7428`

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

=

[ `0.5571` `0.3714` `0.7428` ]

`xx`

`2` `0` `2`

`=2.5997`

`q_2'`

`=a_2-r_(12) * q_1`

=

`2` `0` `2`  -2.5997  `0.5571` `0.3714` `0.7428`

=

`0.5517` `-0.9655` `0.069`

`r_(22)=||q_2'||=sqrt(0.5517^2+(-0.9655)^2+0.069^2)=sqrt(1.2414)=1.1142`

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

=

`1/1.1142 * `

`0.5517` `-0.9655` `0.069`

=

`0.4952` `-0.8666` `0.0619`

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

=

[ `0.5571` `0.3714` `0.7428` ]

`xx`

`4` `2` `3`

`=5.1995`

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

=

[ `0.4952` `-0.8666` `0.0619` ]

`xx`

`4` `2` `3`

`=0.4333`

`q_3'`

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

=

`4` `2` `3`  -5.1995  `0.5571` `0.3714` `0.7428`  -0.4333  `0.4952` `-0.8666` `0.0619`

=

`0.8889` `0.4444` `-0.8889`

`r_(33)=||q_3'||=sqrt(0.8889^2+0.4444^2+(-0.8889)^2)=sqrt(1.7778)=1.3333`

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

=

`1/1.3333 * `

`0.8889` `0.4444` `-0.8889`

=

`0.6667` `0.3333` `-0.6667`

`Q`

`=[q_1,q_2,q_3]`

=

`0.5571` `0.4952` `0.6667` `0.3714` `-0.8666` `0.3333` `0.7428` `0.0619` `-0.6667`

`R`

=

`r_(11)` `r_(12)` `r_(13)` `0` `r_(22)` `r_(23)` `0` `0` `r_(33)`

=

`5.3852` `2.5997` `5.1995` `0` `1.1142` `0.4333` `0` `0` `1.3333`

---

checking `Q xx R = A?`

`Q xx R`

=

`0.5571` `0.4952` `0.6667` `0.3714` `-0.8666` `0.3333` `0.7428` `0.0619` `-0.6667`

`xx`

`5.3852` `2.5997` `5.1995` `0` `1.1142` `0.4333` `0` `0` `1.3333`

=

`3` `2` `4` `2` `0` `2` `4` `2` `3`

and `A`

=

`3` `2` `4` `2` `0` `2` `4` `2` `3`

---