A linear equation is said to be homogeneous when its constant part is zero. For example both of the following are homogeneous:
The following equation, on the other hand, is not homogeneous because its constant part does not equal zero:
In general, a homogeneous equation with variables x1,...,xn, and coefficients a1,...,an looks like:
A homogeneous linear system is on made up entirely of homogeneous equations. For example the following is a homogeneous system
But the following system is not homogeneous because it contains a non-homogeneous equation:
If we write a linear system as a matrix equation, letting A be the coefficient matrix, x the variable vector, and b the known vector of constants, then the equation Ax = b is said to be homogeneous if b is the zero vector.
For example, the following matrix equation is homogeneous
One of the principle advantages to working with homogeneous systems over non-homogeneous systems is that homogeneous systems always have at least one solution, namely, the case where all unknowns are equal to zero. Such a case is called the trivial solution to the homogeneous system.
For instance, looking again at this system:
we see that if x = 0, y = 0, and z = 0, then all three equations are true. This holds equally true for the matrix equation
It is again clear that if all three unknowns are zero, then the equation is true. Whether or not the system has non-trivial solutions is now an interesting question.
It turns out that looking for the existence of non-trivial solutions to matrix equations is closely related to whether or not the matrix is invertible.
Theorem. A square matrix M is invertible if and only if the homogeneous matrix equation Mx=0 does not have any non-trivial solutions.
That is, if Mx=0 has a non-trivial solution, then M is NOT invertible. If, on the other hand, M has an inverse, then Mx=0 only one solution, which is the trivial solution x=0.
Another consequence worth mentioning, we know that if M is a square matrix, then it is invertible only when its determinant |M| is not equal to zero. Hence, Mx=0 will have non-trivial solutions whenever |M| = 0.
Whenever there are fewer equations than there are unknowns, a homogeneous system will always have non-trivial solutions. For example, lets look at the augmented matrix of the above system:
Performing Gauss-Jordan elimination gives us the reduced row echelon form:
Which tells us that z is a free variable, and hence the system has infinitely many solutions.
At this point you might be asking "Why all the fuss over homogeneous systems?". One reason that homogeneous systems are useful and interesting has to do with the relationship to non-homogenous systems. It is often easier to work with the homogenous system, find solutions to it, and then generalize those solutions to the non-homogenous case.
Hence if we are given a matrix equation to solve, and we have already solved the homogeneous case, then we need only find a single particular solution to the equation in order to determine the whole set of solutions.
Furthermore, if the homogeneous case Mx=0 has only the trivial solution, then any other matrix equation Mx=b has only a single solution. We know that this is the case becuase if p=x is a particular solution to Mx=b, then p+h is also a solution where h is a homogeneous solution, and hence p+0 = p is the only solution.