Are Singular Values The Same As Eigenvalues

Are Singular Values the Same as Eigenvalues? A Deep Dive into Matrix Decomposition

This article explores the relationship between singular values and eigenvalues, two fundamental concepts in linear algebra with crucial applications in various fields like machine learning, data analysis, and signal processing. While both relate to the properties of matrices, they are distinct and apply to different scenarios. Understanding their differences is key to effective application.

What are Eigenvalues and Eigenvectors?

Eigenvalues and eigenvectors are intrinsic properties of a square matrix. An eigenvector of a square matrix A is a non-zero vector v that, when multiplied by A, only changes its scale, not its direction. This can be expressed mathematically as:

A v = λ v

where λ (lambda) is the eigenvalue corresponding to the eigenvector v. Eigenvalues represent the scaling factor, indicating how much the eigenvector is stretched or compressed by the transformation represented by matrix A. Finding eigenvalues and eigenvectors involves solving a characteristic equation, which can be computationally intensive for large matrices. They provide crucial information about the matrix's inherent transformations and are used in various applications, including principal component analysis (PCA).

What are Singular Values and Singular Vectors?

Singular Value Decomposition (SVD) is a powerful technique applicable to any matrix, regardless of its shape (square or rectangular). SVD decomposes a matrix A (m x n) into three matrices:

A = U Σ V^T

Where:

• U is an m x m orthogonal matrix whose columns are the left singular vectors.
• Σ is an m x n diagonal matrix containing the singular values (σ₁, σ₂, ..., σᵣ) on its diagonal, where r is the rank of the matrix A. These singular values are always non-negative.
• V^T is the transpose of an n x n orthogonal matrix whose columns are the right singular vectors.

The singular values (σᵢ) represent the scaling factors of the transformation similarly to eigenvalues, but they apply to the stretching and compression of the space spanned by the input and output vectors. The singular vectors (columns of U and V) define the directions of this stretching and compression. SVD provides a geometric interpretation of how the linear transformation represented by the matrix A affects the space.

Key Differences: Why Singular Values are Not the Same as Eigenvalues

The core differences highlight why singular values are not simply a generalization of eigenvalues:

• Matrix Type: Eigenvalues are associated with square matrices, while singular values are defined for any matrix (rectangular or square).
• Decomposition: Eigenvalues are obtained through eigen-decomposition, whereas singular values arise from Singular Value Decomposition (SVD).
• Geometric Interpretation: Eigenvalues represent scaling factors along specific directions (eigenvectors) within the same vector space. Singular values represent scaling factors in the mapping from one vector space (defined by the rows) to another (defined by the columns).
• Non-negativity: Singular values are always non-negative, while eigenvalues can be positive, negative, or zero.
• Relation to the Matrix: Eigenvalues are directly related to the matrix itself through the equation Av = λv. Singular values are related to the product of the matrix and its transpose (A^TA and AA^T), specifically, their eigenvalues. In fact, the singular values of A are the square roots of the eigenvalues of A^TA (or AA^T).

When are they related?

While distinct, there's a connection. The singular values of a matrix are related to the eigenvalues of its associated Gram matrices (A^TA and AA^T). Specifically, the non-zero singular values of a matrix A are the square roots of the non-zero eigenvalues of both A^TA and AA^T. This relationship provides a crucial link between these two concepts. For a symmetric, positive semi-definite matrix, singular values and eigenvalues are identical.

In Conclusion:

Singular values and eigenvalues are powerful tools in linear algebra, providing insights into the properties and transformations of matrices. While related through the square root of eigenvalues of associated Gram matrices, they are fundamentally different. Understanding their distinctions is crucial for appropriately applying them to solve problems across various fields.