A square matrix is one in which the number of rows is equal to the number of columns. A square matrix is the associated matrix of some homogeneous system. Since the matrix is square, the homogeneous system has the same number of equations as there are variables. The matrix is said to be nonsingular if the system has a unique solution. It is said to be singular if the system has an infinite number of solutions. The terms "singular" and "nonsingular" only apply to square matrices.
Note that, by the above theorems, a square matrix is singular if and only it has at least one free variable when it is put into echelon form, which in turn is true if and only if an echelon form of the matrix has at least one row containing only zeros.
I would like to add some geometric perspective to all this, using ideas from the previous section. You should try to develop your higher-dimensional geometric intuition! Of course, if you pick two lines at random, it will be very unlikely that they are identical. The intersection of two planes through the origin is a line, unless the planes happen to be identical. When you add the third plane to the intersection, you are most likely intersecting that plane with a line and the result will be a single point namely the origin , except in the unlikely case that the line happens to lie entirely in the plane.
As the solution set for each equation is added to the intersection, the dimension of the intersection is likely to go down by one. So in this case, we may be as likely to reference only the coefficient matrix and presume that we remember that the final column begins with zeros, and after any number of row operations is still zero. Then the system has infinitely many solutions. One has a unique solution, while the other has infinitely many. The set of solutions to a homogeneous system which by Theorem HSC is never empty is of enough interest to warrant its own name.
However, we define it as a property of the coefficient matrix, not as a property of some system of equations. In the Archetypes Archetypes each example that is a system of equations also has a corresponding homogeneous system of equations listed, and several sample solutions are given.
These solutions will be elements of the null space of the coefficient matrix. We will look at one example. Section HSE Homogeneous Systems of Equations In this section we specialize to systems of linear equations where every equation has a zero as its constant term.
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. 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.
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.
Sign in. Sign In. Homogeneous Linear Systems. Author: c o.
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