Tagiuraoob

2022-02-22

What I wanted to ask was , given a homogeneous system of n variables , like this having 4 variables:
${a}_{1}x+{a}_{2}y+{a}_{3}z+{a}_{4}w=0$
${a}_{2}x+{a}_{3}y+{a}_{4}z+{a}_{1}w=0$
${a}_{3}x+{a}_{4}y+{a}_{1}z+{a}_{2}w=0$
${a}_{4}x+{a}_{1}y+{a}_{2}z+{a}_{3}w=0$
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
$\left({a}_{1},{a}_{2},{a}_{3},{a}_{4}\right)$

junoon363km

General sollution to AX=0 is the kernel Ker(A).
To say AX=0 only have zero solution (trivial kernel $K\text{er}\left(A\right)=\left\{0\right\}$) is equivalent to verify the row reduced echelon form of matrix A:
$rrefg\left(A\right)=I$
$\left(A=\left(\begin{array}{cccc}{a}_{1}& {a}_{2}& {a}_{3}& {a}_{4}\\ {a}_{2}& {a}_{3}& {a}_{4}& {a}_{1}\\ {a}_{3}& {a}_{4}& {a}_{1}& {a}_{2}\\ {a}_{4}& {a}_{1}& {a}_{2}& {a}_{3}\end{array}\right)\right)$
Just need to verify:
$\left(rref\left(A\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\right)$
For second question, treat ${a}_{i}$ as unknowns:
$x{a}_{1}+y{a}_{2}+z{a}_{3}+w{a}_{4}=0$
$w{a}_{1}+x{a}_{2}+y{a}_{3}+z{a}_{4}=0$
$z{a}_{1}+w{a}_{2}+x{a}_{3}+w{a}_{4}=0$
$y{a}_{1}+z{a}_{2}+w{a}_{3}+x{a}_{4}=0$
Then it is another linear system you can solve.

Do you have a similar question?