Decomposition Of A Full Column Rank Matrix Into An Orthogonal Matrix And A Unit Triangular Matrix Is Unique

Introduction

In linear algebra, matrix decomposition is a crucial technique used to break down a matrix into simpler components. One such decomposition is the QR decomposition, which expresses a matrix as the product of an orthogonal matrix and a unit upper triangular matrix. In this article, we will explore the uniqueness of this decomposition for a full column rank matrix.

What is a Full Column Rank Matrix?

A matrix $A_{m \times k}$ is said to have full column rank if its columns are linearly independent. This means that none of the columns can be expressed as a linear combination of the other columns. In other words, the columns of the matrix form a basis for the column space of the matrix.

QR Decomposition

The QR decomposition is a factorization of a matrix $A_{m \times k}$ into the product of an orthogonal matrix $Q$ and a unit upper triangular matrix $R$. The decomposition is given by:

$$A = QR$$

where $Q$ is an orthogonal matrix, and $R$ is a unit upper triangular matrix.

Uniqueness of QR Decomposition

The book that I am reading states that the QR decomposition of a full column rank matrix is unique. To understand why this is the case, let's consider the following:

Suppose we have two QR decompositions of the same matrix $A$:

$$A = QR_1$$

$$A = QR_2$$

where $Q$ is the same orthogonal matrix in both decompositions, and $R_1$ and $R_2$ are two different unit upper triangular matrices.

Subtracting the two equations, we get:

$$0 = R_1Q - QR_2$$

Since $Q$ is an orthogonal matrix, we can multiply both sides of the equation by $Q^T$ to get:

$$0 = Q^TR_1Q - QR_2Q^T$$

Since $Q^TQ = I$, we can simplify the equation to:

$$0 = R_1 - QR_2Q^T$$

Now, let's consider the product $QR_2Q^T$. Since $Q$ is an orthogonal matrix, we know that $Q^TQ = I$. Therefore, we can simplify the product to:

$$QR_2Q^T = R_2$$

Substituting this back into the previous equation, we get:

$$0 = R_1 - R_2$$

This implies that $R_1 = R_2$, which means that the QR decomposition of a full column rank matrix is unique.

Proof of Uniqueness

To prove the uniqueness of the QR decomposition, we need to show that if $A = QR_1$ and $A = QR_2$ are two QR decompositions of the same matrix $A$, then $R_1 = R_2$.

Let's consider the following:

$$QR_1 = QR_2$$

Multiplying both sides of the equation by $Q^T$, we get:

$$Q^TQR_1 = Q^TQR_2$$

Since $Q^TQ = I$, we can simplify the equation to:

$$R_1 = R_2$$

This shows that if A = QR_ and $A = QR_2$ are two QR decompositions of the same matrix $A$, then $R_1 = R_2$, which means that the QR decomposition of a full column rank matrix is unique.

Conclusion

In this article, we explored the uniqueness of the QR decomposition of a full column rank matrix. We showed that if $A = QR_1$ and $A = QR_2$ are two QR decompositions of the same matrix $A$, then $R_1 = R_2$, which means that the QR decomposition of a full column rank matrix is unique.

References

• [1] Golub, G. H., & Van Loan, C. F. (2013). Matrix computations. Johns Hopkins University Press.
• [2] Strang, G. (2016). Linear algebra and its applications. Brooks Cole.
• [3] Trefethen, L. N., & Bau, D. (1997). Numerical linear algebra. SIAM.

Further Reading

• QR decomposition: A tutorial by the MathWorks
• QR decomposition: A tutorial by Wolfram MathWorld
• QR decomposition: A tutorial by MIT OpenCourseWare

Image Credits

• Image 1: QR decomposition by the MathWorks
• Image 2: QR decomposition by Wolfram MathWorld
• Image 3: QR decomposition by MIT OpenCourseWare
  Frequently Asked Questions (FAQs) about QR Decomposition ===========================================================

Q: What is QR decomposition?

A: QR decomposition is a factorization of a matrix $A_{m \times k}$ into the product of an orthogonal matrix $Q$ and a unit upper triangular matrix $R$. The decomposition is given by:

$$A = QR$$

where $Q$ is an orthogonal matrix, and $R$ is a unit upper triangular matrix.

Q: What is the purpose of QR decomposition?

A: QR decomposition is used in various applications such as:

• Linear least squares problems
• Linear regression analysis
• Signal processing
• Image processing
• Machine learning

Q: What is the difference between QR decomposition and other matrix decompositions?

A: QR decomposition is different from other matrix decompositions such as:

• LU decomposition: LU decomposition is a factorization of a matrix into the product of a lower triangular matrix and an upper triangular matrix.
• Cholesky decomposition: Cholesky decomposition is a factorization of a symmetric positive definite matrix into the product of a lower triangular matrix and its transpose.
• Singular value decomposition (SVD): SVD is a factorization of a matrix into the product of three matrices: a unitary matrix, a diagonal matrix, and another unitary matrix.

Q: How is QR decomposition used in linear least squares problems?

A: QR decomposition is used in linear least squares problems to find the best fit line or curve to a set of data points. The QR decomposition is used to decompose the matrix of coefficients into the product of an orthogonal matrix and a unit upper triangular matrix. The solution to the linear least squares problem is then found by solving a system of linear equations involving the unit upper triangular matrix.

Q: How is QR decomposition used in signal processing?

A: QR decomposition is used in signal processing to decompose a signal into its constituent parts. The QR decomposition is used to decompose the matrix of coefficients into the product of an orthogonal matrix and a unit upper triangular matrix. The solution to the signal processing problem is then found by solving a system of linear equations involving the unit upper triangular matrix.

Q: What are the advantages of QR decomposition?

A: The advantages of QR decomposition are:

• It is a stable algorithm for solving linear least squares problems.
• It is a fast algorithm for solving linear least squares problems.
• It is a robust algorithm for solving linear least squares problems.

Q: What are the disadvantages of QR decomposition?

A: The disadvantages of QR decomposition are:

• It requires a large amount of memory to store the orthogonal matrix and the unit upper triangular matrix.
• It requires a large amount of computational resources to perform the QR decomposition.

Q: How is QR decomposition implemented in practice?

A: QR decomposition is implemented in practice using various algorithms such as:

• Gram-Schmidt process
• Householder transformation
• Givens rotation

Q: What are the applications of QR decomposition?

A: The applications of QR decomposition are:

• Linear least squares problems
• Linear regression analysis
• Signal processing
• Image processing
• Machine learning

Q: What are the limitations of QR decomposition?

A: The limitations of QR decomposition are:

• It is not suitable for solving ill-conditioned linear systems.
• It is not suitable for solving linear systems with a large number of variables.

Q: How can QR decomposition be used in machine learning?

A: QR decomposition can be used in machine learning to:

• Solve linear least squares problems
• Solve linear regression problems
• Perform dimensionality reduction
• Perform feature extraction

Q: What are the benefits of using QR decomposition in machine learning?

A: The benefits of using QR decomposition in machine learning are:

• It is a fast and efficient algorithm for solving linear least squares problems.
• It is a robust algorithm for solving linear least squares problems.
• It is a stable algorithm for solving linear least squares problems.

Q: What are the challenges of using QR decomposition in machine learning?

A: The challenges of using QR decomposition in machine learning are:

• It requires a large amount of memory to store the orthogonal matrix and the unit upper triangular matrix.
• It requires a large amount of computational resources to perform the QR decomposition.

Q: How can QR decomposition be used in image processing?

A: QR decomposition can be used in image processing to:

• Perform image denoising
• Perform image deblurring
• Perform image segmentation
• Perform image feature extraction

Q: What are the benefits of using QR decomposition in image processing?

A: The benefits of using QR decomposition in image processing are:

• It is a fast and efficient algorithm for solving linear least squares problems.
• It is a robust algorithm for solving linear least squares problems.
• It is a stable algorithm for solving linear least squares problems.

Q: What are the challenges of using QR decomposition in image processing?

A: The challenges of using QR decomposition in image processing are:

• It requires a large amount of memory to store the orthogonal matrix and the unit upper triangular matrix.
• It requires a large amount of computational resources to perform the QR decomposition.