The Givens rotation matrix (or plane rotation matrix) is an orthogonal matrix that is often used to transform a real matrix into an equivalent one, typically by annihilating the entries below its main diagonal.
Here is a definition.
Definition
Let
and
be two real numbers such that
.
Let
be the
identity matrix. Let
and
be two integers such that
.
The Givens rotation matrix
is
the
matrix whose entries are all equal to the corresponding entries of
,
except
for
Let us immediately see some examples.
Example
The following is a
Givens
matrix:
In
this case,
,
and
.
Thus, the matrix is obtained by modifying the second and fourth rows of a
identity matrix.
Example
Another example is
which
has been obtained by modifying the first and fourth rows of a
identity matrix.
Example
A smaller rotation matrix
follows:
Givens matrices are orthogonal (i.e., their columns are orthonormal).
Proposition
A Givens matrix
is orthogonal, that
is,
Let
We
need to prove that
By
using the definition of
matrix product, we can see that the latter equation holds if and only
if
where
denotes the
-th
row of
.
We are going to prove that this is true. When
and
,
then
because
the rows of the identity matrix are orthonormal. When
,
then
When
,
then
When
and
(or
and
),
so that
,
then we
have
Finally,
when either
or
is equal to one of
or
(and the other one is not equal to
or
),
we have
either
or
which
completes the proof.
Let
be
a
Givens rotation matrix.
Let
be a
matrix.
What happens when we compute the
productthat
is, when we use
to perform an equivalent transformation on
?
By the usual interpretation of
matrix
products as linear combinations, we can see that the product is a new
matrix whose rows are all equal to the corresponding rows of
,
except for the
-th
and
-th.
In particular, if
and
,
then
But we
have
Example
Let
and
Then,
the product is
Suppose that, as in the previous section, we perform an equivalent
transformation of a
matrix
by using a Givens rotation
.
Further suppose that
.
How can we set
in such a way that the transformation annihilates the entry
?
We already know that, after the transformation, the
-th
row
is
Therefore,
In order to have
,
we must
have
But we must also satisfy the
constraint
These two equations are solved
by
Example
Define
Let
us find a rotation matrix that allows us to annihilate the entry
.
First of all, we need a non-zero entry to use as a pivot. We choose
.
Thus, the rows involved in the rotation are the first and the third one. As a
consequence, our Givens matrix has the
form
The
numbers
and
are chosen as
follows:
Thus,
the transformation
is
which
achieves the desired annihilation.
Below you can find some exercises with explained solutions.
Define the
matrixFind
a Givens rotation matrix that transforms
into an upper triangular
matrix.
We need to annihilate the entry
.
We can do so by pivoting on
.
Thus,
and
.
Moreover,
Therefore,
the rotation matrix
is
The
equivalent transformation
is
Please cite as:
Taboga, Marco (2021). "Givens rotation matrix", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/Givens-rotation.
Most of the learning materials found on this website are now available in a traditional textbook format.