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Left Singular Vectors
Columns of the orthogonal matrix U in the SVD decomposition, forming an orthonormal basis of the starting space and corresponding to the eigenvectors of AA^T.
Right Singular Vectors
Columns of the orthogonal matrix V in the SVD decomposition, forming an orthonormal basis of the arrival space and corresponding to the eigenvectors of A^TA.
Orthogonal Matrix
Square matrix whose columns and rows are orthogonal unit vectors, satisfying the property Q^TQ = QQ^T = I, where I is the identity matrix.
Numerical Rank
Number of singular values above a certain tolerance threshold, determining the effective rank of a matrix in a numerical context where very small values are considered zero.
SVD Truncation
Dimensionality reduction technique consisting of keeping only the k largest singular values and their associated vectors, creating a rank-k approximation of the original matrix.
Complex SVD
Extension of the SVD decomposition to matrices with complex coefficients, where matrices U and V become unitary (U^*U = I) and Σ contains real non-negative singular values.
Moore-Penrose Pseudoinverse
Generalization of the matrix inverse for non-square or singular matrices, efficiently computed via the SVD decomposition as A^+ = VΣ^+U^T, where Σ^+ is obtained by inverting the non-zero singular values.
Frobenius Norm
Matrix norm defined as the square root of the sum of the squares of all elements, equivalent to the square root of the sum of the squares of the singular values in the context of SVD decomposition.
Norm 2 (or Spectral Norm)
Matrix norm induced by the vector Euclidean norm, equal to the largest singular value of the matrix and measuring its maximum amplification on a unit vector.
Matrix Condition Number
Ratio between the largest and smallest nonzero singular values, measuring the sensitivity of the solution of a linear system to perturbations in the data, with a high condition number indicating an ill-conditioned matrix.
Incremental SVD
Algorithm for updating the SVD decomposition when new columns or rows are added to a matrix, avoiding a complete recalculation and particularly useful for continuous data streams.
Randomized SVD
Probabilistic method that accelerates the computation of SVD decomposition for very large matrices by using random projections to capture the dominant subspace before computing the exact SVD on this approximation.
Eckart-Young Theorem
Theoretical foundation guaranteeing that the best rank-k approximation of a matrix (in terms of the 2-norm or Frobenius norm) is obtained by truncating its SVD decomposition to the k largest singular values.
Thick SVD
Variant of the SVD decomposition that computes more singular values than the theoretical rank of the matrix, useful for capturing noise structure or for applications in robust principal component analysis.
Thin SVD
Economical form of the SVD decomposition where the U and V matrices contain only the columns corresponding to nonzero singular values, reducing storage and computational complexity.
Bi-orthogonal Decomposition
Alternative to SVD for non-normal matrices, decomposing a matrix A into XBY^T where X and Y are invertible matrices and B is bidiagonal, serving as an intermediate step in some SVD computation algorithms.