Givens rotation matrix

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.

Table of contents

Definition
Orthogonality
Equivalent transformations
Annihilation of entries
Solved exercises
1. Exercise 1

Definition

Here is a definition.

Definition Let and be two real numbers such that $c^{2}+s^{2}=1$ . 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 [eq5]

Let us immediately see some examples.

Example The following is a Givens matrix: [eq6] In this case, , and . Thus, the matrix is obtained by modifying the second and fourth rows of a identity matrix.

Example Another example is [eq7] which has been obtained by modifying the first and fourth rows of a identity matrix.

Example A smaller rotation matrix follows: [eq8]

Orthogonality

Givens matrices are orthogonal (i.e., their columns are orthonormal).

Proposition A Givens matrix is orthogonal, that is,

Proof

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 [eq12] where $G_{kbullet }$ denotes the -th row of . We are going to prove that this is true. When and , then [eq13] because the rows of the identity matrix are orthonormal. When , thenWhen , thenWhen and (or and ), so that , then we haveFinally, when either or is equal to one of or (and the other one is not equal to or ), we have eitherorwhich completes the proof.

Equivalent transformations

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 [eq22]

Example Let [eq23] and [eq24] Then, the product is [eq25]

Annihilation of entries

Suppose that, as in the previous section, we perform an equivalent transformation of a matrix by using a Givens rotation .

Further suppose that $A_{mm} eq 0$ .

How can we set in such a way that the transformation annihilates the entry $A_{nm}$ ?

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 [eq32]

Example Define [eq33] Let us find a rotation matrix that allows us to annihilate the entry $A_{31}=4$ . First of all, we need a non-zero entry to use as a pivot. We choose $A_{11}=3$ . Thus, the rows involved in the rotation are the first and the third one. As a consequence, our Givens matrix has the form [eq34] The numbers and are chosen as follows: [eq35] Thus, the transformation is [eq36] which achieves the desired annihilation.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Define the matrix [eq37] Find a Givens rotation matrix that transforms into an upper triangular matrix.

Solution

We need to annihilate the entry $A_{32}=-3$ . We can do so by pivoting on $A_{22}=4$ . Thus, and . Moreover, [eq38] Therefore, the rotation matrix is [eq39] The equivalent transformation is [eq40]

How to cite

Please cite as:

Taboga, Marco (2021). "Givens rotation matrix", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/Givens-rotation.