Prove The Diagonal Entries Of R, In Any QR Decomposition, Are Non-zero ; Hence R Is Invertible.

Prove the Diagonal Entries of R are Non-Zero in QR Decomposition

In linear algebra, the QR decomposition is a factorization of a matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R. This decomposition is particularly useful in solving systems of linear equations and in many applications of linear algebra. However, one of the key properties of the QR decomposition is that the diagonal entries of the upper triangular matrix R are non-zero, which implies that R is invertible. In this article, we will prove this property and explore its implications.

The QR decomposition of a matrix A is given by:

A = QR

where Q is an orthogonal matrix and R is an upper triangular matrix. The orthogonal matrix Q satisfies the property:

Q^T Q = I

where I is the identity matrix. The upper triangular matrix R has non-zero diagonal entries, which we will prove in this article.

To prove that the diagonal entries of R are non-zero, we will use the Gram-Schmidt process to construct the orthogonal matrix Q. The Gram-Schmidt process is a method for orthonormalizing a set of vectors. We will take a set of vectors $\mathbf{u_j}$ and use the Gram-Schmidt process to construct the orthonormal vectors $\mathbf{q_j}$.

Let $\mathbf{u_j}$ be a set of vectors in $\mathbb{R}^n$. We will use the Gram-Schmidt process to construct the orthonormal vectors $\mathbf{q_j}$ as follows:

$$\mathbf{q_1} = \frac{\mathbf{u_1}}{\|\mathbf{u_1}\|}$$

$$\mathbf{q_j} = \frac{\mathbf{u_j} - \sum_{i=1}^{j-1} \langle \mathbf{u_j} , \mathbf{q_i} \rangle \mathbf{q_i}}{\|\mathbf{u_j} - \sum_{i=1}^{j-1} \langle \mathbf{u_j} , \mathbf{q_i} \rangle \mathbf{q_i}\|}$$

where $\langle \mathbf{u_j} , \mathbf{q_i} \rangle$ is the inner product of $\mathbf{u_j}$ and $\mathbf{q_i}$.

Now, let's consider the diagonal entry $r_{jj}$ of the matrix R. We can write:

$$r_{jj} = \langle \mathbf{q_j} , \mathbf{q_j} \rangle$$

Using the definition of $\mathbf{q_j}$, we can write:

$$\mathbf{q_j} = \frac{\mathbf{u_j} - \sum_{i=1}^{j-1} \langle \mathbf{u_j} , \mathbf{q_i} \rangle \mathbf{q_i}}{\|\mathbf{u_j} - \sum_{i=1}^{j-1} \langle \mathbf{u_j} , \mathbf{q_i} \rangle \mathbf{q_i}\|}$$

Now, let's consider the inner product $\langle \mathbf{u_j} , \mathbf{q_i} \rangle$. We can write:

$$\langle \mathbf{u_j} , \mathbf{q_i} \rangle = \langle \mathbf{u_j} , \mathbf{u_i} - \sum_{k=1}^{i-1} \langle \mathbf{u_i} , \mathbf{q_k} \rangle \mathbf{q_k} \rangle$$

Using the linearity of the inner product, we can write:

$$\langle \mathbf{u_j} , \mathbf{q_i} \rangle = \langle \mathbf{u_j} , \mathbf{u_i} \rangle - \sum_{k=1}^{i-1} \langle \mathbf{u_j} , \mathbf{q_k} \rangle \langle \mathbf{u_i} , \mathbf{q_k} \rangle$$

Now, let's consider the term $\langle \mathbf{u_j} , \mathbf{q_k} \rangle$. We can write:

$$\langle \mathbf{u_j} , \mathbf{q_k} \rangle = \frac{\langle \mathbf{u_j} , \mathbf{u_k} \rangle}{\|\mathbf{u_k}\|}$$

Using the definition of $\mathbf{q_k}$, we can write:

$$\mathbf{q_k} = \frac{\mathbf{u_k}}{\|\mathbf{u_k}\|}$$

Now, let's consider the term $\langle \mathbf{u_i} , \mathbf{q_k} \rangle$. We can write:

$$\langle \mathbf{u_i} , \mathbf{q_k} \rangle = \frac{\langle \mathbf{u_i} , \mathbf{u_k} \rangle}{\|\mathbf{u_k}\|}$$

Now, let's substitute these expressions back into the expression for $\langle \mathbf{u_j} , \mathbf{q_i} \rangle$. We get:

$$\langle \mathbf{u_j} , \mathbf{q_i} \rangle = \langle \mathbf{u_j} , \mathbf{u_i} \rangle - \sum_{k=1}^{i-1} \frac{\langle \mathbf{u_j} , \mathbf{u_k} \rangle}{\|\mathbf{u_k}\|} \frac{\langle \mathbf{u_i} , \mathbf{u_k} \rangle}{\|\mathbf{u_k}\|}$$

Now, let's consider the term $\langle \mathbf{u_j} , \mathbf{u_i} \rangle$. We can write:

$$\langle \mathbf{u_j} , \mathbf{u_i} \rangle = \|\mathbf{u_j}\| \|\mathbf{u_i}\| \cos \theta_{ji}$$

where $\theta_{ji}$ is the angle between $\mathbf{u_j}$ and $\mathbf{u_i}$.

Now, let's substitute this expression back into the expression for $\langle \mathbf{u_j} , \mathbf{q_i} \rangle$. We get:

$$\langle \mathbf{u_j} , \mathbf{q_i} \rangle = \|\mathbf{u_j}\| \|\mathbf{u_i}\| \cos \theta_{ji} - \sum_{k=1}^{i-1} \frac{\langle \mathbf{u_j} , \mathbf{u_k} \rangle}{\|\mathbf{u_k}\|} \frac{\langle \mathbf{u_i} , \mathbf{u_k} \rangle}{\|\mathbf{u_k}\|}$$

Now, let's consider the term $\|\mathbf{u_j}\| \|\mathbf{u_i}\| \cos \theta_{ji}$. We can write:

$$\|\mathbf{u_j}\| \|\mathbf{u_i}\| \cos \theta_{ji} = \|\mathbf{u_j}\| \|\mathbf{u_i}\| \left( \frac{\langle \mathbf{u_j} , \mathbf{u_i} \rangle}{\|\mathbf{u_j}\| \|\mathbf{u_i}\|} \right)$$

Now, let's substitute this expression back into the expression for $\langle \mathbf{u_j} , \mathbf{q_i} \rangle$. We get:

$$\langle \mathbf{u_j} , \mathbf{q_i} \rangle = \langle \mathbf{u_j} , \mathbf{u_i} \rangle - \sum_{k=1}^{i-1} \frac{\langle \mathbf{u_j} , \mathbf{u_k} \rangle}{\|\mathbf{u_k}\|} \frac{\langle \mathbf{u_i} , \mathbf{u_k} \rangle}{\|\mathbf{u_k}\|}$$

Now, let's consider the term $\langle \mathbf{u_j} , \mathbf{u_i} \rangle$. We can write:

$$\langle \mathbf{u_j} , \mathbf{u_i} \rangle = \|\mathbf{u_j}\| \|\mathbf{u_i}\| \cos \theta_{ji}$$

Now, let's substitute this expression back into the expression for $\langle \mathbf{u_j} , \mathbf{q_i} \rangle$. We get:

\langle \mathbf{u_j} , \mathbf{q_i} \rangle = \|\mathbf{u_j}\| \|\mathbf{u_i}\| \cos \theta_{ji} - \sum_{k=1}^{i-1} \frac{\langle \mathbf{u_j} , \mathbf{u_k} \rangle}{\|\mathbf{u_k}\|} \frac<br/> **Q&A: Prove the Diagonal Entries of R are Non-Zero in QR Decomposition**

In our previous article, we proved that the diagonal entries of the upper triangular matrix R in the QR decomposition are non-zero, which implies that R is invertible. In this article, we will answer some frequently asked questions related to this topic.

A: The QR decomposition is a factorization of a matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R. This decomposition is particularly useful in solving systems of linear equations and in many applications of linear algebra.

A: The QR decomposition is useful because it allows us to solve systems of linear equations efficiently. By decomposing the matrix A into Q and R, we can solve the system of linear equations Ax = b by solving the system Rx = Q^T b, where x is the solution vector.

A: The diagonal entries of R being non-zero is significant because it implies that R is invertible. This is important because it allows us to solve the system of linear equations Rx = Q^T b efficiently.

A: We can prove that the diagonal entries of R are non-zero by using the Gram-Schmidt process to construct the orthogonal matrix Q. The Gram-Schmidt process is a method for orthonormalizing a set of vectors. By using this process, we can show that the diagonal entries of R are non-zero.

A: The Gram-Schmidt process is a method for orthonormalizing a set of vectors. It is a way of constructing an orthonormal basis for a vector space. The process involves taking a set of vectors and using them to construct a new set of orthonormal vectors.

A: The Gram-Schmidt process is used to construct the orthogonal matrix Q in the QR decomposition. By using the Gram-Schmidt process, we can construct Q and R such that A = QR.

A: The QR decomposition has many applications in linear algebra and beyond. Some examples include:

• Solving systems of linear equations
• Finding the eigenvalues and eigenvectors of a matrix
• Computing the singular value decomposition (SVD) of a matrix
• Image and signal processing

A: There are many ways to implement the QR decomposition in practice. Some common methods include:

• Using a library or package that implements the QR decomposition, such as LAPACK or NumPy
• Writing your own implementation of the QR decomposition using the Gram-Schmidt process
• Using a QR decomposition algorithm, such as the Householder transformation or the Givens rotation

In this article, we have answered some frequently asked questions related to the QR decomposition and the proof that the diagonal entries of R are non-zero. We have also discussed some applications of the QR decomposition and provided some information on how to implement it in practice.