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Problem: lq decomposition [[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]] [ Calculator, Method and examples ]

Solution:
Your problem -> lq decomposition [[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]

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)=4.2426

q_1 = 1/(||q_1'||) * q_1' = 1/4.2426 *
 1 -1 4
=
 0.2357 -0.2357 0.9428

r_(12)=q_1^T * a_2 =
 [ 0.2357 -0.2357 0.9428 ]
xx
 1 4 -2
=-2.5927

q_2'=a_2-r_(12) * q_1 =
 1 4 -2
+2.5927
 0.2357 -0.2357 0.9428
=
 29/18 61/18 4/9

r_(22)=||q_2'||=sqrt(1.6111^2+3.3889^2+0.4444^2)=sqrt(14.2778)=3.7786

q_2 = 1/(||q_2'||) * q_2' = 1/3.7786 *
 29/18 61/18 4/9
=
 0.4264 0.8969 0.1176

r_(13)=q_1^T * a_3 =
 [ 0.2357 -0.2357 0.9428 ]
xx
 1 4 2
=1.1785

r_(23)=q_2^T * a_3 =
 [ 0.4264 0.8969 0.1176 ]
xx
 1 4 2
=4.2491

q_3'=a_3-r_(13) * q_1-r_(23) * q_2 =
 1 4 2
-1.1785
 0.2357 -0.2357 0.9428
-4.2491
 0.4264 0.8969 0.1176
=
 -1.0895 0.4669 0.3891

r_(33)=||q_3'||=sqrt((-1.0895)^2+0.4669^2+0.3891^2)=sqrt(1.5564)=1.2476

q_3 = 1/(||q_3'||) * q_3' = 1/1.2476 *
 -1.0895 0.4669 0.3891
=
 -0.8733 0.3743 0.3119

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