A homogeneous system of equations is a system in which the vector of constants on the right-hand side of the equals sign is zero.
In this lecture we provide a general characterization of the set of solutions of a homogeneous system.
A homogenous system has the
formwhere
is a
matrix of coefficients,
is a
vector of unknowns and
is the
zero vector.
Example
The
systemwhich
can be written in matrix form
as
is
homogeneous.
By performing elementary row operations on a homogenous system, we obtain equivalent systems that are all homogenous. In fact, elementary row operations (multiplying an equation by a non-zero constant; adding a multiple of one equation to another equation; interchanging two equations) leave the zero vector of constants on the right-hand side of the equals sign unaffected.
As a consequence, we can transform the original system into an equivalent
homogeneous
systemwhere
the matrix
is in row echelon form (REF).
A homogeneous system always has the
solutionwhich
is called trivial solution.
Remember that the columns of a REF matrix are of two kinds:
basic columns: they contain a pivot (i.e., a non-zero entry such that we find only zero entries in the quadrant starting from the pivot and extending below it and to its left);
non-basic columns: they do not contain a pivot.
Example
Consider the following
matrix in row echelon
form:
The
first and the third columns are basic, while the second and the fourth are
non-basic.
Suppose that the
REF matrix
has
basic columns.
Without loss of generality, we can assume that the first
columns are basic and the last
are non-basic (we can re-number the unknowns if necessary).
Partition the matrix
into two
blocks:
where
is the
sub-matrix of basic columns and
is the
sub-matrix of non-basic columns.
Similarly, partition the vector of unknowns into two
blocks:where
is the
vector of basic variables and
is the
vector of non-basic variables.
Then, we can write the system of equations
asor
The general solution of the homogeneous
systemis
the set of all possible solutions, that is, the set of all
that satisfy the system of equations.
We already know that, if the system has a solution, then we can arbitrarily
choose the values of the non-basic variables
and then find, by the back-substitution algorithm, the values of the basic
variables
that solve the system. In the homogeneous case, the existence of a solution is
not an issue because the vector of constants is zero (revise the lecture on
the row echelon form if you
are wondering why).
Therefore, there is a unique
that solves equation (1) for any arbitrary choice of
.
Since
is full-rank and
,
the matrix
is full-rank (see the lecture on the
rank of matrix
products). Therefore, we can pre-multiply equation (1) by
so as to
obtain
Define
Then, we
have
The latter can be used to characterize the general solution of the homogeneous system: it explicitly links the values of the basic variables to those of the non-basic variables that can be set arbitrarily.
Denote by
the general solution (i.e., the set of all possible solutions). Then, we
have
The product
can be seen as a
linear
combination of the columns of
whose coefficients are the non-basic
variables:
Thus, each column of
is a particular solution of the system, obtained by setting its corresponding
non-basic variable equal to
and all the other non-basic variables equal to
.
By taking linear combination of these particular solutions, we obtain the
general solution.
Clearly, the general solution embeds also the trivial one, which is obtained by setting all the non-basic variables to zero.
Example
Consider the homogeneous
systemwhere
and
Then,
we can
define
The
system can be written
as
but
since
is the identity matrix, we
have
Thus,
the general solution of the system is the set of all vectors
that
satisfy
Below you can find some exercises with explained solutions.
Define the
matrix
Find the general solution of the
systemwhere
is a
vector of unknowns.
The matrix
is not in row echelon form, but we can subtract three times the first row from
the third one in order to obtain an equivalent matrix in row echelon
form:
Thus,
we can discuss the solutions of the equivalent
system
Since
both of the two columns of
are basic, there are no unknowns to choose arbitrarily. As a consequence, the
only solution of the system is the trivial one
(
).
Define the
matrix
Find the general solution of the
systemwhere
is a
vector of unknowns.
For convenience, we are going to
transform
into a reduced row echelon
form matrix. We divide the second row by
;
then, we subtract two times the second row from the first one. The result is
an equivalent matrix in reduced row echelon
form:
We
can now discuss the solutions of the equivalent
system
The
system can be written
as
Thus,
the general solution of the system is the set of all vectors
that
satisfy
Please cite as:
Taboga, Marco (2021). "Homogeneous system", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/homogeneous-system.
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