Properties of Orthogonal Matrices

Oct 13, 2025 by thesweetdesigns.com 116 views

Introduction

In the realm of Linear Algebra, matrices play a crucial role in solving systems of equations, finding the least squares solution, and many other applications. When dealing with rectangular matrices, it's essential to understand the properties and relationships between them. In this article, we will explore a fundamental lemma that relates two rectangular matrices $A$ and $B$ , and its implications on the structure of matrix $A$ . Specifically, we will investigate the conditions under which $A$ can be expressed as a product of two matrices, $U$ and $B$ , where $U$ is an orthogonal matrix.

The Lemma

The lemma states that if we have two rectangular matrices $A$ and $B$ , both of size $n \times m$ , such that:

A^T A = B^T B

where $A^T$ and $B^T$ are the transpose of matrices $A$ and $B$ , respectively, then we can express matrix $A$ as a product of two matrices, $U$ and $B$ , as follows:

A = UB

where $U$ is an orthogonal matrix, i.e., $U^T U = I$ , where $I$ is the identity matrix.

Proof of the Lemma

To prove this lemma, we will start by expressing matrix $A$ as a product of two matrices, $U$ and $B$ . Let's assume that:

A = UB

where $U$ is an orthogonal matrix. Taking the transpose of both sides, we get:

A^T = B^T U^T

Now, let's multiply both sides of the equation by $A$ :

A^T A = B^T U^T A

Since $U^T U = I$ , we can simplify the right-hand side of the equation:

A^T A = B^T B

This shows that the condition $A^T A = B^T B$ is satisfied.

Implications of the Lemma

The lemma has several important implications. First, it shows that if we have two rectangular matrices $A$ and $B$ that satisfy the condition $A^T A = B^T B$ , then we can express matrix $A$ as a product of two matrices, $U$ and $B$ , where $U$ is an orthogonal matrix. This provides a useful decomposition of matrix $A$ into two simpler matrices.

Second, the lemma implies that the columns of matrix $A$ can be expressed as a linear combination of the columns of matrix $B$ . This is because the product $UB$ represents a linear transformation that maps the columns of $B$ to the columns of $A$ .

Properties of Orthogonal Matrices

An orthogonal matrix $U$ has several important properties. First, it satisfies the condition $U^T U = I$ , which means that the columns of $U$ are orthonormal vectors. Second, the determinant of an orthogonal matrix is either $1$ or $-1$ . Finally, the inverse of an orthogonal matrix is its transpose, i.e., $U^{-1} = U^T$ .

Applications of the Lemma

The lemma has several applications in Linear Algebra and other fields. One of the most important applications is in the context of Singular Value Decomposition (SVD). SVD is a factorization technique that decomposes a matrix into three matrices: $U$ , $Σ$ , and $V^T$ , where $U$ and $V$ are orthogonal matrices, and $Σ$ is a diagonal matrix containing the singular values of the original matrix.

The lemma is used in the proof of the SVD theorem, which states that any matrix $A$ can be decomposed into the product $U Σ V^T$ , where $U$ and $V$ are orthogonal matrices, and $Σ$ is a diagonal matrix containing the singular values of $A$ .

Conclusion

In conclusion, the lemma states that if two rectangular matrices $A$ and $B$ satisfy the condition $A^T A = B^T B$ , then we can express matrix $A$ as a product of two matrices, $U$ and $B$ , where $U$ is an orthogonal matrix. This provides a useful decomposition of matrix $A$ into two simpler matrices and has several important implications, including the fact that the columns of matrix $A$ can be expressed as a linear combination of the columns of matrix $B$ . The lemma has several applications in Linear Algebra and other fields, including the context of Singular Value Decomposition (SVD).

References

[1] Strang, G. (1988). Linear Algebra and Its Applications. 2nd ed. San Diego: Harcourt Brace Jovanovich.
[2] Golub, G. H., & Van Loan, C. F. (1996). Matrix Computations. 3rd ed. Baltimore: Johns Hopkins University Press.
[3] Horn, R. A., & Johnson, C. R. (1991). Matrix Analysis. Cambridge: Cambridge University Press.
Q&A: If $A^TA = B^T B$ then $A = UB$ with $U^T U = I$ ===========================================================

Q: What is the significance of the lemma $A^TA = B^T B$ ?

A: The lemma $A^TA = B^T B$ is significant because it provides a condition under which two rectangular matrices $A$ and $B$ can be related to each other. Specifically, it shows that if $A^TA = B^T B$ , then we can express matrix $A$ as a product of two matrices, $U$ and $B$ , where $U$ is an orthogonal matrix.

Q: What is the relationship between the columns of matrices $A$ and $B$ ?

A: The lemma implies that the columns of matrix $A$ can be expressed as a linear combination of the columns of matrix $B$ . This is because the product $UB$ represents a linear transformation that maps the columns of $B$ to the columns of $A$ .

Q: What are the properties of an orthogonal matrix $U$ ?

A: An orthogonal matrix $U$ has several important properties. First, it satisfies the condition $U^T U = I$ , which means that the columns of $U$ are orthonormal vectors. Second, the determinant of an orthogonal matrix is either $1$ or $-1$ . Finally, the inverse of an orthogonal matrix is its transpose, i.e., $U^{-1} = U^T$ .

Q: How is the lemma used in the context of Singular Value Decomposition (SVD)?

A: The lemma is used in the proof of the SVD theorem, which states that any matrix $A$ can be decomposed into the product $U Σ V^T$ , where $U$ and $V$ are orthogonal matrices, and $Σ$ is a diagonal matrix containing the singular values of $A$ . The lemma is used to show that the columns of matrix $A$ can be expressed as a linear combination of the columns of matrix $V$ .

Q: What are the implications of the lemma for the columns of matrix $A$ ?

A: The lemma implies that the columns of matrix $A$ can be expressed as a linear combination of the columns of matrix $B$ . This means that the columns of $A$ can be written as a linear combination of the columns of $B$ , with coefficients given by the columns of matrix $U$ .

Q: Can the lemma be used to find the inverse of matrix $A$ ?

A: No, the lemma cannot be used to find the inverse of matrix $A$ . The lemma only provides a relationship between the columns of matrices $A$ and $B$ , and does not provide a way to find the inverse of $A$ .

Q: Are there any other applications of the lemma?

A: Yes, the lemma has several other applications in Linear Algebra and other fields. For example, it can be used to find the least squares solution to a system of linear equations, and to solve optimization problems.

Q: How can the lemma be used to solve optimization problems?

A: The lemma can be used to solve optimization problems by expressing the objective function as a quadratic form, and then using the lemma to find the minimum or maximum of the quadratic form. This is a powerful technique that can be used to solve a wide range of optimization problems.

Q: Are there any limitations to the lemma?

A: Yes, there are several limitations to the lemma. For example, it only applies to rectangular matrices, and does not provide a way to find the inverse of matrix $A$ . Additionally, the lemma assumes that the matrices $A$ and $B$ are real-valued, and does not provide a way to extend the result to complex-valued matrices.

Q: Can the lemma be used to find the eigenvalues and eigenvectors of matrix $A$ ?

A: No, the lemma cannot be used to find the eigenvalues and eigenvectors of matrix $A$ . The lemma only provides a relationship between the columns of matrices $A$ and $B$ , and does not provide a way to find the eigenvalues and eigenvectors of $A$ .

Q: Are there any other related results that are similar to the lemma?

A: Yes, there are several other related results that are similar to the lemma. For example, the Gram-Schmidt process is a technique that can be used to find an orthonormal basis for a vector space, and the QR decomposition is a technique that can be used to decompose a matrix into a product of an orthogonal matrix and an upper triangular matrix.