How to solve a matrix multiplication that results in zero? F*x = 0
I would like to solve a system of linear equations, all of which equal to zero. Is there a way to do this using linear algebra without decomposing into separate equations.
The equation is simple, F is a known matrix, lets say a 3x3
x is a vector of unknown variables, for example, x,y,z
0 is a zero vector, in this example 0,0,0
The equation is F*x = 0
What is the best approach to solving this?
If I did inv(F)*F*x = inv(F)*0 the answer I get is x = 0, y = 0, z = 0
Whats the best approach to solving this problem using matrices? Like adding an identity matrix to both sides?
$\endgroup$ 12 Answers
$\begingroup$Consider the equation $A\textbf{x} = \textbf{0}$ with $A$ an invertible matrix. Then clearly you can multiply the inverse matrix to both sides and get
$$ A^{-1}A\textbf{x} = A^{-1}\textbf{0} \implies \textbf{x} = \textbf{0} $$
Thus, the zero vector is the unique solution to the equation $A\textbf{x} = \textbf{0}$. So, if your matrix is invertible, then you have already solved the problem.
Assuming that the matrix $A$ is not invertible, then the above manipulation does not work, because $A^{-1}$ doesn't exist. To solve the equation in this case you can write down the augmented matrix $\left [ A \ | \ \textbf{0} \right ]$ and perform Gaussian elimination to find all solutions.
$\endgroup$ $\begingroup$The solutions to the equation $F x = 0$ constitute the null space or kernel of the linear operator corresponding to the matrix $F$. A standard way of finding it is Gaussian elimination.
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