In singular value decomposition, which statement is true about Σ?

• A. The diagonal Σ contains the singular values.
• B. U is the identity matrix.
• C. V^T contains the left singular vectors.
• D. Σ is the original data matrix.

Answer: (A)

In singular value decomposition, any matrix M is written as M = U Σ V^T, where U and V are orthogonal and Σ is diagonal with nonnegative entries. Those diagonal entries are the singular values, placed on Σ’s diagonal in descending order. They quantify how much M stretches along each pair of left and right singular directions. The left singular vectors are the columns of U, and the right singular vectors are the columns of V (so V^T has those right singular vectors as its rows). Σ is not the original data matrix; it’s the scaling part that holds the singular values. So the statement that the diagonal of Σ contains the singular values is the correct one.