What I wanted to ask was , given a homogeneous system of n variables , like this having 4 variables:

Tagiuraoob

Tagiuraoob

Answered question

2022-02-22

What I wanted to ask was , given a homogeneous system of n variables , like this having 4 variables:
a1x+a2y+a3z+a4w=0
a2x+a3y+a4z+a1w=0
a3x+a4y+a1z+a2w=0
a4x+a1y+a2z+a3w=0
Here , as we know that this system has zero solution and we can see that the coefficients are rotating in each of the linear equation .
So , is there any general form for solution of this system other than the zero solution ?
Also , what if we are given the non-zero solution then can we find the values of
(a1,a2,a3,a4)

Answer & Explanation

junoon363km

junoon363km

Beginner2022-02-23Added 8 answers

General sollution to AX=0 is the kernel Ker(A).
To say AX=0 only have zero solution (trivial kernel Ker(A)={0}) is equivalent to verify the row reduced echelon form of matrix A:
rrefg(A)=I
In your case:
(A=(a1a2a3a4a2a3a4a1a3a4a1a2a4a1a2a3))
Just need to verify:
(rref(A)=(1000010000100001))
For second question, treat ai as unknowns:
xa1+ya2+za3+wa4=0
wa1+xa2+ya3+za4=0
za1+wa2+xa3+wa4=0
ya1+za2+wa3+xa4=0
Then it is another linear system you can solve.

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