What is QR factorization?

Decomposing an $m \times n$ matrix $A$ with linearly independent column vectors into two matrices $Q$ and $R$, where $A = QR$, is known as QR factorization.

• $Q$ is an $m \times n$ matrix that has orthonormalAll vectors in the set S are orthogonal to each other and have a magnitude of 1. column vectors.

• $R$ is an $n \times n$ upper triangularAll the elements below the main diagonal are zero in an upper triangular matrix., invertible matrix with non-zero diagonal elements.

Procedure

Let's consider $A$ to be a $3 \times 3$ matrix with the following definition:

The matrix can be decomposed into the following two matrices:

Here, $q_{1}, q_{2}, q_{3}$ are orthonormal vectors calculated using the Gram-Schmidt process.

$R$ will be calculated using the vector dot product between the column vectors of matrix $A$ and matrix $Q$.

Example

Given that the following matrix $A$ has linearly independent column vectors, we can find its $QR$ factorization in the following way:

$Q$ can be calculated using the Gram-Schmidt process:

Note: To learn about Gram-Schmidt process in detail, click here.

$R$ is calculated as individual column vectors to depict the process in detail. Here are the steps of this process:

Step 1

The column vector $r_{1}$ can be calculated as follows:

Step 2

The column vector $r_{2}$ can be calculated as follows:

Step 3

The column vector $r_{3}$ can be calculated as follows:

Step 4

The column vectors $r_{1}, r_{2}, r_{3}$ give the following matrix $R$:

Solution

Matrix $A$ has been factorized into two matrices, $Q$ and $R$ , using QR decomposition:

The calculated matrices follow the equation of the QR decomposition:

Quiz

To test your understanding of the QR concept, attempt the following questions:

Applications

QR decomposition makes calculations easier and is used in numerous applications, such as the following:

• It reduces the order of matrices.

• It speeds up matrix operations.

• It is used in successive cancellation detection in MIMO systems.

• It is used in optimal ordered detection to maximize the signal-to-noise ratio.