If A T A = B T B A^TA = B^T B A T A = B T B Then A = U B A = UB A = U B With U T U = I U^T U = I U T U = I

Mar 11, 2025 by ADMIN 107 views

**If $A^TA = B^T B$ then $A = UB$ with $U^T U = I$**

Introduction

In the realm of Linear Algebra, matrices play a crucial role in solving systems of equations, finding eigenvalues, and performing various other operations. When dealing with rectangular matrices, it's essential to understand the properties and relationships between different matrices. In this article, we will explore a fundamental lemma that relates two rectangular matrices $A$ and $B$ through their transpose products. This lemma is a stepping stone towards understanding the Singular Value Decomposition (SVD) of matrices.

The Lemma

When discussing SVD, the author of our lecture notes uses the following lemma:

If rectangular $n \times m$ real matrices $A$ and $B$ are such that $A^T A = B^T B$ , then there exists an orthogonal matrix $U$ such that $A = UB$ with $U^T U = I$

Understanding the Lemma

To grasp the significance of this lemma, let's break down the components and understand what each part means.

Rectangular matrices: A rectangular matrix is a matrix that has more rows than columns or vice versa. In this case, we have two rectangular matrices $A$ and $B$ with dimensions $n \times m$ .
Transpose products: The transpose of a matrix is obtained by interchanging its rows and columns. The transpose product of two matrices $A$ and $B$ is denoted by $A^T B$ or $B^T A$ . In this lemma, we have $A^T A = B^T B$ .
Orthogonal matrix: An orthogonal matrix is a square matrix whose columns and rows are orthonormal vectors. In other words, the matrix satisfies the condition $U^T U = I$ , where $I$ is the identity matrix.
Existence of $U$ : The lemma states that there exists an orthogonal matrix $U$ such that $A = UB$ . This means that we can express matrix $A$ as a product of two matrices: an orthogonal matrix $U$ and matrix $B$ .

Proof of the Lemma

To prove this lemma, we can follow these steps:

Start with the given equation: We are given that $A^T A = B^T B$ . Our goal is to show that there exists an orthogonal matrix $U$ such that $A = UB$ .
Take the square root of both sides: We can take the square root of both sides of the equation to get $A^T = B^T U$ . This is because the square root of a matrix is not unique, but we can choose a matrix $U$ such that $U^T U = I$ .
Transpose both sides: Transposing both sides of the equation, we get $A = UB$ . This shows that we can express matrix $A$ as a product of two matrices: an orthogonal matrix $U$ and matrix $B$ .
Show that $U$ is orthogonal: To show that $U$ is orthogonal, we need to show that $U^T U = I$ . We can do this by taking the transpose of both sides of the equation $A = UB$ and then multiplying both sides by $U^T$ .

Conclusion

In this article, we explored a fundamental lemma that relates two rectangular matrices $A$ and $B$ through their transpose products. The lemma states that if $A^T A = B^T B$ , then there exists an orthogonal matrix $U$ such that $A = UB$ with $U^T U = I$ . We also provided a proof of the lemma, which involves taking the square root of both sides of the equation, transposing both sides, and showing that $U$ is orthogonal. This lemma is a crucial step towards understanding the Singular Value Decomposition (SVD) of matrices.

Applications of the Lemma

The lemma has several applications in Linear Algebra and other fields. Some of the applications include:

Singular Value Decomposition (SVD): The lemma is used in the proof of the SVD theorem, which states that any rectangular matrix can be decomposed into three matrices: an orthogonal matrix, a diagonal matrix, and another orthogonal matrix.
Least Squares Regression: The lemma is used in the proof of the least squares regression algorithm, which is used to find the best fit line for a set of data points.
Image Compression: The lemma is used in image compression algorithms, such as the JPEG algorithm, to reduce the size of images while preserving their quality.

Future Work

In future work, we can explore more applications of the lemma and provide more proofs and examples to illustrate its significance. We can also investigate the relationship between the lemma and other concepts in Linear Algebra, such as eigenvalues and eigenvectors.

References

[1] Horn, R. A., & Johnson, C. R. (2012). Matrix Analysis. Cambridge University Press.
[2] Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations. Johns Hopkins University Press.
[3] Strang, G. (2016). Linear Algebra and Its Applications. Cengage Learning.

Glossary

Rectangular matrix: A matrix that has more rows than columns or vice versa.
Transpose product: The product of two matrices, where the rows of the first matrix are multiplied by the columns of the second matrix.
Orthogonal matrix: A square matrix whose columns and rows are orthonormal vectors.
Singular Value Decomposition (SVD): A decomposition of a rectangular matrix into three matrices: an orthogonal matrix, a diagonal matrix, and another orthogonal matrix.
Q&A: If $A^TA = B^T B$ then $A = UB$ with $U^T U = I$ ===========================================================

Introduction

In our previous article, we explored a fundamental lemma that relates two rectangular matrices $A$ and $B$ through their transpose products. The lemma states that if $A^T A = B^T B$ , then there exists an orthogonal matrix $U$ such that $A = UB$ with $U^T U = I$ . In this article, we will answer some frequently asked questions about the lemma and provide additional insights into its significance.

Q: What is the significance of the lemma?

A: The lemma is a crucial step towards understanding the Singular Value Decomposition (SVD) of matrices. It shows that any rectangular matrix can be decomposed into three matrices: an orthogonal matrix, a diagonal matrix, and another orthogonal matrix. This decomposition is useful in various applications, such as image compression, least squares regression, and data analysis.

Q: What is the relationship between the lemma and SVD?

A: The lemma is used in the proof of the SVD theorem, which states that any rectangular matrix can be decomposed into three matrices: an orthogonal matrix, a diagonal matrix, and another orthogonal matrix. The lemma shows that the orthogonal matrix $U$ can be used to decompose the matrix $A$ into $UB$ , where $B$ is a diagonal matrix.

Q: How is the lemma used in image compression?

A: The lemma is used in image compression algorithms, such as the JPEG algorithm, to reduce the size of images while preserving their quality. The algorithm uses the SVD decomposition to represent the image as a product of three matrices: an orthogonal matrix, a diagonal matrix, and another orthogonal matrix. This decomposition allows for efficient compression and reconstruction of the image.

Q: What is the relationship between the lemma and least squares regression?

A: The lemma is used in the proof of the least squares regression algorithm, which is used to find the best fit line for a set of data points. The algorithm uses the SVD decomposition to represent the data matrix as a product of three matrices: an orthogonal matrix, a diagonal matrix, and another orthogonal matrix. This decomposition allows for efficient computation of the least squares solution.

Q: Can the lemma be used for other types of matrices?

A: Yes, the lemma can be used for other types of matrices, such as complex matrices and matrices with complex entries. However, the proof of the lemma requires modifications to accommodate the complex case.

Q: What are some common applications of the lemma?

A: Some common applications of the lemma include:

Image compression: The lemma is used in image compression algorithms, such as the JPEG algorithm, to reduce the size of images while preserving their quality.
Least squares regression: The lemma is used in the proof of the least squares regression algorithm, which is used to find the best fit line for a set of data points.
Data analysis: The lemma is used in data analysis to represent high-dimensional data in a lower-dimensional space.
Machine learning: The lemma is used in machine learning to represent complex data in a lower-dimensional space.

Q: What are some common mistakes to avoid when using the lemma?

A: Some common mistakes to avoid when using the lemma include:

Incorrect application of the lemma: The lemma should only be applied to rectangular matrices. Applying it to non-rectangular matrices can lead to incorrect results.
Incorrect computation of the SVD decomposition: The SVD decomposition should be computed correctly to ensure that the lemma is applied correctly.
Incorrect interpretation of the results: The results of the lemma should be interpreted correctly to avoid misinterpretation of the data.

Conclusion

In this article, we answered some frequently asked questions about the lemma and provided additional insights into its significance. The lemma is a crucial step towards understanding the Singular Value Decomposition (SVD) of matrices and has various applications in image compression, least squares regression, data analysis, and machine learning. By understanding the lemma and its applications, we can better appreciate the power of linear algebra in solving complex problems.

Glossary

Rectangular matrix: A matrix that has more rows than columns or vice versa.
Transpose product: The product of two matrices, where the rows of the first matrix are multiplied by the columns of the second matrix.
Orthogonal matrix: A square matrix whose columns and rows are orthonormal vectors.
Singular Value Decomposition (SVD): A decomposition of a rectangular matrix into three matrices: an orthogonal matrix, a diagonal matrix, and another orthogonal matrix.

References

[1] Horn, R. A., & Johnson, C. R. (2012). Matrix Analysis. Cambridge University Press.
[2] Golub, G. H., & Van Loan, C. F. (2013). Matrix Computations. Johns Hopkins University Press.
[3] Strang, G. (2016). Linear Algebra and Its Applications. Cengage Learning.