A matrix polynomial is a linear combination of the powers of a square matrix.
Remember that the powers of a square
matrix
are obtained by multiplying
by itself several times.
In particular, given a positive integer
and a
matrix
,
we
have
A commonly adopted convention is that the
-th
power of
is the
identity matrix:
We can now define matrix polynomials.
Definition
Let
be a
matrix. Let
be a non-negative integer. We say that
is a polynomial in
of degree
if and only
if
where
is the
identity matrix,
are scalars, and
.
Thus,
is a matrix having the same dimension as
,
obtained as a linear
combination of powers of
.
The scalars
are the so-called coefficients of the matrix polynomial.
In the above definition
is assumed to be a non-negative integer. If
for any matrix
(i.e., the matrix polynomial is identically equal to the zero matrix), then we
adopt the convention that the degree of
is
.
Example
Let
be a square matrix.
Then,
is
a matrix polynomial in
of degree
.
Note that a matrix polynomial as defined above is not an
ordinary
polynomial. In fact, in an ordinary polynomial, the elements being raised
to powers and their coefficients are required to come from the same
field. Instead, in a matrix
polynomial, the coefficients come from a field (the so-called field of
scalars) while the matrices being raised to powers come from a different set,
which is not even a field. For example, in a field the multiplication
operation must be commutative, but
matrix multiplication is
not commutative. Actually, the set of
matrices, for fixed
,
is a ring, an algebraic structure satisfying a weaker set of axioms than that
satisfied by a field.
Nonetheless, if
is a field, such as the set of real numbers
or the set of complex numbers
,
and
is an ordinary
polynomial
then
we can use
to define, by extension, an associated matrix
polynomial
provided
that the entries of the matrix
belong to the field
.
It is common practice to switch back and forth between these two kinds of polynomials. For example, we often: 1) write a matrix polynomial; 2) derive its associated ordinary polynomial; 2) use the theory of ordinary polynomials to write the polynomial in a different form (e.g., we factorize it); 3) use the new form of the ordinary polynomial (e.g., its factorization) to rewrite the original matrix polynomial.
Example
Define the matrix
polynomialIts
associated ordinary polynomial
is
which
can be re-written
as
Then,
we
have
In theory, every time that we switch back and forth between the two kinds of polynomials, we should check whether the properties of ordinary polynomials that we are using hold also for matrix polynomials.
In practice, if we revise previous lectures on ordinary polynomials (e.g., polynomial division and greatest common divisors), we will realize that basically every definition, proof and proposition in those lectures is valid also for matrix polynomials. The reason is that, even if we lose some properties of fields when we deal with matrices, we do not need those properties to manipulate polynomials. Moreover, we can even use the commutative property of products, as explained in the next section.
We have already said that, in general, matrix multiplication is not
commutative. However, the multiplication of two polynomials
and
in the same matrix
is
commutative:
This perhaps obvious fact (a proof of which can be found in a solved exercise at the end of this lecture) is extremely useful, and it is used to prove important results in linear algebra (e.g., the result below on null spaces).
We can also define polynomials in linear operators, but, as we will shortly argue, we do not need to develop a separate theory because (as long as we deal with finite-dimensional vector spaces) a polynomial in a linear operator is always associated to an analogous matrix polynomial and we can study the properties of the latter.
Definition
Let
be a vector space and
a linear operator. Let
be a non-negative integer. We say that
is a polynomial in
of degree
if and only
if
where
are scalars,
,
is the identity operator that associates each member of
to itself, and
denotes the operator obtained by composing
times
:
Let
be a basis for the
vector space
.
Remember that, provided
is finite-dimensional, any linear operator
is associated to a square matrix, called matrix of the linear operator with
respect to
and denoted by
such that, for any
,
where
and
respectively denote the
coordinate vectors of
and
with respect to
.
Moreover, the
composition of
operators can be performed by multiplying their respective
matrices:
Therefore, a polynomial
such as the one defined above is an operator whose matrix is a matrix
polynomial in
.
Thus, as we have said at the beginning of this section, we do not need a separate theory for operator polynomials and we can deal with them by using polynomials in the respective matrices.
A simple albeit often used result follows.
Proposition
Let
be the space of
vectors. Let
be a
matrix. Let
be a polynomial in
.
Then, the null space
of
is an invariant subspace of
under the linear transformation defined by
.
Denote by
the null space of
.
Choose any
.
We have
Thus,
,
which proves that
is invariant under
.
Once we know the eigenvalues of
,
we can easily compute the eigenvalues of
.
Proposition
Let
be a
matrix. Let
be a polynomial in
.
Let
be an eigenvalue of
.
Then,
is an eigenvalue of
.
We have
previously
demonstrated that, if
is an eigenvalue of
associated to the eigenvector
,
then
is an eigenvalue of
associated to the same eigenvector
.
Suppose that
Choose
any eigenvalue
of
and an associated eigenvector
.
Then,
which
proves the proposition.
A frequently used concept is that of an annihilating polynomial.
Definition
Let
be a
matrix. Let
be a polynomial in
.
We say that
is an annihilating polynomial if and only
if
In the lecture on the Cayley-Hamilton theorem, we will carefully explain one of the most important results in the theory of matrix polynomials, which states that the characteristic polynomial is an annihilating polynomial.
Below you can find some exercises with explained solutions.
Let
be a square matrix. Transform the ordinary
polynomial
into
a polynomial in
.
The matrix polynomial
is
Let
be a square matrix. Transform the ordinary
polynomial
into
a polynomial in
.
The matrix polynomial
is
Prove that the multiplication of two polynomials
and
in the same matrix
is
commutative:
Suppose
thatand
Then
Please cite as:
Taboga, Marco (2021). "Matrix polynomial", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/matrix-polynomial.
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