I want to solve a system of linear equation by LU decomposition method (Crout matrix decomposition). Most importantly, I want to take all elements of the principal diagonal of U as 1. 

How to do that?

Equations: 4x+y+z=4, x+4y-2z=4, 3x+2y-4z=6. 

I want to simplify the attached expression by applying the relation \( k_1 + k_2 + k_3 = 0 \).

For instance, the terms \( k_3 + k_1 - k_2 \) become \(-2k_2\) and \( k_1 + k_2 - k_3 \) become \(-2k_3\), etc.

Instead of manually changing each term, I am looking for a systematic method to apply this relation to simplify the given expression.

aabb.mw

How to simplify the expression where each vectors k_i and q has real components, i.e., etc

Only W_i, U_i,j, V_i,j are complex numbers, so conjugate (bar) is applicable for them. How to make declaration, so that all other bars are removed and simplify the expression.

aa.mw

I want to write  in maple using summation notation for symbolic calculation. The output should come as 

W_1exp(k_1.q)+W_2exp(k_2.q)+W_3exp(k_3.q)

where k_j.q are the dot product of the vectors k_j and q. 

mp.mw

How to do this. Please help. The summation is not working for me, only suffix k is coming  

Can I solve the given biquadratic equation in terms of sigma. I only need real positive root, if any

6125*_Z^4 + 68644*_Z^3*sigma - 219625*_Z^3 + 255712*_Z^2*sigma - 959250*_Z^2 + 238144*_Z*sigma - 1113500*_Z - 245000

I tried with solve for variable but it does not work.