In singular value decomposition, which statement is true?

In singular value decomposition, which statement is true?

• A. The matrix A is decomposed as U V^T, and Σ contains singular values.
• B. A = U Σ V^T; Σ contains left singular vectors.
• C. Σ contains eigenvalues.
• D. A = U Σ V^T; Σ contains singular values.

Answer: (D)

In SVD, any matrix A can be written as A = U Σ V^T, where U and V are orthogonal and Σ is a diagonal (or rectangular diagonal) matrix of nonnegative numbers. The columns of U are the left singular vectors, the columns of V are the right singular vectors, and the diagonal entries of Σ are the singular values, typically arranged in decreasing order. The singular values satisfy σ_i^2 = eigenvalues of A^T A (and of A A^T). So the statement A = U Σ V^T; Σ contains singular values is the correct one. The middle matrix Σ is essential, left singular vectors live in U, and eigenvalues are not the entries of Σ (they relate to A^T A).

In SVD, any matrix A can be written as A = U Σ V^T, where U and V are orthogonal and Σ is a diagonal (or rectangular diagonal) matrix of nonnegative numbers. The columns of U are the left singular vectors, the columns of V are the right singular vectors, and the diagonal entries of Σ are the singular values, typically arranged in decreasing order. The singular values satisfy σ_i^2 = eigenvalues of A^T A (and of A A^T).

So the statement A = U Σ V^T; Σ contains singular values is the correct one. The middle matrix Σ is essential, left singular vectors live in U, and eigenvalues are not the entries of Σ (they relate to A^T A).