Singular value decomposition (SVD) is a powerful matrix factorization technique that extends eigendecomposition to any matrix. It breaks down matrices into simpler components, revealing crucial structural information and enabling various applications in data analysis and engineering.
SVD connects to spectral theory by generalizing the spectral theorem to non-square and non-normal matrices. This versatility makes SVD a fundamental tool for understanding matrix properties, solving complex problems, and uncovering hidden patterns in data across diverse fields.
Singular Value Decomposition of Matrices
Definition and Structure
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Singular value decomposition (SVD) factorizes real or complex matrices, generalizing eigendecomposition to any m × n matrix
For m × n matrix A, orthogonal matrices U and V and diagonal matrix Σ exist such that A = UΣV^T
U dimensions m × m
Σ dimensions m × n
V^T dimensions n × n
Diagonal entries σ_i of Σ called singular values, arranged in descending order
Columns of U termed left singular vectors
Columns of V termed right singular vectors
SVD exists for all matrices (square, rectangular, invertible, non-invertible)
Number of non-zero singular values equals matrix A's rank
SVD decomposes matrix into simpler matrices, revealing structural information (low-rank approximations, matrix norms)
Mathematical Properties
SVD always exists, providing a universal matrix decomposition method
Uniqueness of SVD components
Singular values are unique
Singular vectors may have sign ambiguity for non-zero singular values
Singular vectors corresponding to repeated singular values may form an orthonormal basis
Relationship to other matrix properties
Frobenius norm: ∥A∥F=∑i=1min(m,n)σi2
Nuclear norm: ∥A∥∗=∑i=1min(m,n)σi
Spectral norm: ∥A∥2=σ1 (largest singular value)
Connection to matrix rank
Rank(A) = number of non-zero singular values
Low-rank approximation obtained by truncating smaller singular values
Components of the SVD
Computation and Interpretation
Calculate SVD by finding eigenvectors and eigenvalues of A^T A and AA^T
Right singular vectors (V columns) derived from A^T A eigenvectors
Left singular vectors (U columns) derived from AA^T eigenvectors
Singular values σ_i calculated as square roots of non-zero eigenvalues of both A^T A and AA^T
Construct Σ by placing singular values on diagonal in descending order