Understanding the Distinction Between Singular Values and Eigenvalues in Decomposition

In the realm of linear algebra, singular value decomposition (SVD) and eigenvalues serve as fundamental tools for analyzing matrices. Despite their importance, a common question that arises is why singular value decomposition (SVD) is not considered a generalization of eigendecomposition, and why we use the square roots of the eigenvalues of ATA or AAT. In this article, we will explore these concepts and clarify the reasons behind the choice of square roots over the eigenvalues themselves.

Misconception: Singular Value Decomposition as a Generalization of Eigendecomposition

Many students and practitioners often mistakenly believe that singular value decomposition (SVD) is a generalization of eigendecomposition. However, this is not the case. While both concepts are pivotal in linear algebra, they serve different purposes and have distinct properties. One of the key differences is that eigenvalues can be complex, whereas singular values are always non-negative real numbers. This fundamental property sets singular values apart from eigenvalues, making them better suited for certain applications such as data compression and image processing.

The Role of ATA and AAT in SVD

In the context of SVD, the matrices ATA and AAT play a crucial role. They are symmetric and therefore possess all the properties of eigenvalues and eigenvectors. Specifically, the eigenvalues of ATA or AAT correspond to the squares of the singular values of A, and their eigenvectors correspond to the right and left singular vectors of A, respectively.

The use of squares in the context of ATA or AAT is not arbitrary. By squaring the singular values, we capture the essence of the lengths of the semi-principal axes of the data represented by the matrix A. This is directly related to the intuitive understanding of the geometric transformation properties of the matrix A. Squared values ensure that the dimensions of the transformed space are consistent and maintain the physical quantities such as length, time, and mass.

Why Use the Square Roots of Eigenvalues?

Given that the eigenvalues of ATA or AAT are the squares of the singular values, it might seem logical to use the eigenvalues themselves in SVD. However, there are compelling reasons to prefer the square roots of these eigenvalues:

Preserving Dimensionality: The singular values of a matrix A represent the magnitudes of the semi-principal axes in the transformed space. By taking the square roots of the eigenvalues of ATA or AAT, we ensure that these magnitudes are preserved in the original physical dimensions. This is particularly important in applications where maintaining the physical interpretation of the data is crucial. Data Interpretation: The singular values provide a more direct and interpretable measure of the data. For example, in image processing, the singular values can be directly related to the brightness or contrast of the image. Using the square roots of the eigenvalues allows for a more intuitive understanding of the data's characteristics. Practical Applications: In many practical applications, such as principal component analysis (PCA) and data compression, the square roots of the eigenvalues are used to represent the variances along the principal components. This choice simplifies the interpretation and manipulation of the data.

Intuitive Explanation and Mathematical Insight

A comprehensive understanding of the relationship between singular values and eigenvalues can be gained by considering the intuitive interpretation provided in the Wikipedia article on singular value decomposition. The article explains that the squares of the singular values are equivalent to the eigenvalues of ATA or AAT. This equivalence highlights the geometric transformation properties of the matrix A and the scaling of the semi-principal axes.

Mathematically, if A is an m x n matrix with singular values σ1, σ2, ..., σmin(m,n), then the eigenvalues of ATA or AAT are precisely the squares of these singular values: σ12, σ22, ..., σmin(m,n)2. Taking the square roots of these eigenvalues yields the singular values themselves, providing a direct and meaningful representation of the matrix A's action on the data.

Conclusion

In conclusion, while singular value decomposition and eigenvalues are both essential tools in linear algebra, they serve distinct purposes. The choice of using the square roots of the eigenvalues of ATA or AAT in SVD is driven by practical considerations such as preserving dimensionality, ensuring intuitive data interpretation, and facilitating practical applications. Understanding this distinction enriches our knowledge of these fundamental concepts and their applications in various fields.