We now consider how to represent the
inverse of a linear map.
We start by recalling some facts about function
Some functions have no inverse, or have an inverse on the left side
or right side only.
is the projection map
and is the embedding
the composition is the identity map on .
We say is a left inverse map
of or, what is the same thing,
that is a right inverse map
However, composition in the other order
doesn't give the identity map— here is a vector that is not
sent to itself under .
In fact, the projection
has no left inverse at all.
For, if were to be a left inverse of
then we would have
for all of the infinitely many 's. But no function can send a single argument to more than one value.
(An example of a function with no inverse on either side
is the zero transformation on .)
Some functions have a
two-sided inverse map, another function
that is the inverse of the first, both from the left and from the right.
For instance, the map given by
has the two-sided inverse
In this subsection we will focus on two-sided inverses.
The appendix shows that a function
has a two-sided inverse if and only if it is both one-to-one and onto.
The appendix also shows that if a function has a two-sided inverse then
it is unique, and so it is called
"the" inverse, and is denoted .
So our purpose in this subsection is, where a linear map has an inverse,
to find the relationship between and
(recall that we have shown, in Theorem II.2.21
of Section II of this chapter, that if a linear map has an inverse
then the inverse is a linear map also).
A matrix is a left inverse matrix of the matrix if is the identity matrix. It is a right inverse matrix if is the identity. A matrix with a two-sided inverse is an invertible matrix. That two-sided inverse is called the inverse matrix and is denoted .
Because of the correspondence between linear maps and matrices,
statements about map inverses translate into statements about matrix inverses.
If a matrix has both a left inverse and a right inverse then the two are equal.
A matrix is invertible if and only if it is nonsingular.
(For both results.) Given a matrix , fix spaces of appropriate dimension for the domain and codomain. Fix bases for these spaces. With respect to these bases, represents a map . The statements are true about the map and therefore they are true about the matrix.
A product of invertible matrices is invertible— if and are invertible and if is defined then is invertible and .
(This is just like the prior proof except that it requires two maps.)
Fix appropriate spaces and bases and consider the represented maps and
Note that is a two-sided map inverse of since
This equality is reflected in the matrices representing the maps, as required.
Here is the arrow diagram giving the relationship
between map inverses and matrix inverses.
It is a special case
of the diagram for function composition and matrix multiplication.
Beyond its place in our general program of
seeing how to represent map operations,
another reason for our interest in inverses comes from solving
A linear system is equivalent to a matrix equation, as here.
By fixing spaces and bases (e.g., and ),
we take the matrix to represent some map .
Then solving the system is the same as
asking: what domain vector is mapped by to the result
If we could invert then we could solve the system
to get .
We can find a left inverse for the matrix just given
by using Gauss' method to solve the resulting linear system.
Answer: , , , and .
This matrix is actually the two-sided inverse of ,
as can easily be checked.
With it we can solve the system () above by
applying the inverse.
Why solve systems this way, when
Gauss' method takes less arithmetic
(this assertion can be made precise by counting the
number of arithmetic operations,
as computer algorithm designers do)?
Beyond its conceptual appeal of fitting into our program of
discovering how to represent the various map operations,
solving linear systems by using the matrix inverse has
at least two advantages.
First, once the work of finding an inverse has been done,
solving a system with the
same coefficients but different constants is easy and fast: if
we change the entries on the right of the system ()
then we get a related problem
with a related solution method.
In applications, solving many systems having the same matrix of
coefficients is common.
Another advantage of inverses is that we can
explore a system's sensitivity to changes in the constants.
For example, tweaking the on the right of the system () to
can be solved with the inverse.
to show that changes by of the tweak while moves by of that tweak. This sort of analysis is used, for example, to decide how accurately data must be specified in a linear model to ensure that the solution has a desired accuracy.
We finish by describing the computational procedure
usually used to find the inverse matrix.
A matrix is invertible if and only if it can be written as the product of elementary reduction matrices. The inverse can be computed by applying to the identity matrix the same row steps, in the same order, as are used to Gauss-Jordan reduce the invertible matrix.
A matrix is invertible if and only if it is nonsingular and thus
Gauss-Jordan reduces to the identity.
By Corollary 3.22 this reduction can
be done with elementary matrices
This equation gives the two halves of the result.
First, elementary matrices are invertible and their inverses are also
Applying to the left of both sides of that equation, then
, etc., gives as the product of
(the is here to cover the trivial case).
Second, matrix inverses are unique and so comparison of the above equation with shows that . Therefore, applying to the identity, followed by , etc., yields the inverse of .
To find the inverse of
we do Gauss-Jordan reduction, meanwhile performing the same operations on
For clerical convenience we write the matrix and the identity side-by-side,
and do the reduction steps together.
This calculation has found the inverse.
This one happens to start with a row swap.
A non-invertible matrix is detected by the fact that the left half won't
reduce to the identity.
This procedure will find the inverse of a general matrix.
The case is handy.
We have seen here, as in the Mechanics of Matrix Multiplication subsection,
that we can exploit the correspondence between
linear maps and matrices.
So we can fruitfully study both maps and matrices, translating back and forth
to whichever helps us the most.
Over the entire four subsections of
this section we have developed an algebra system for matrices.
We can compare it with the familiar algebra system for the real numbers.
Here we are working not with numbers but with matrices.
We have matrix addition and subtraction operations,
and they work in much the same
way as the real number operations, except that they only combine same-sized
We also have a matrix multiplication operation
and an operation inverse to multiplication.
These are somewhat like the familiar real number operations
(associativity, and distributivity over addition, for example), but
there are differences (failure of commutativity, for example).
And, we have scalar multiplication, which is in some ways another extension
of real number multiplication.
This matrix system provides an example that algebra
systems other than the
elementary one can be interesting and useful.