A retail team holds a large matrix with one row per store and one column per product. The team wants to factor that matrix into orthogonal pieces plus a diagonal that records how strong each piece is, then keep the strongest pieces. Which method is that factorization?
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Short Explanation
Pulling a chord into separate notes, each with its own loudness, is the shape of this job. You want singular value decomposition: orthogonal pieces of the matrix plus a diagonal of strengths. The trap is a neighbor map or a cluster label that never factors the matrix.
Full Explanation
Singular value decomposition factors a matrix into orthogonal pieces and a diagonal whose entries record the strength of each piece. The store-by-product matrix is the object that factorization acts on, and keeping the strongest diagonal entries is a later choice about how much of the factorization to retain. A two-dimensional neighbor map places similar stores near each other for display and does not return orthogonal factors or a diagonal of strengths. Density clustering labels tight groups and sparse stores and does not factor the matrix into pieces. A nearest-neighbor vote copies a class onto a new store from nearby labeled stores; it classifies one row and does not decompose the matrix. Exam caveat: the method asked for is the factorization itself, and how many leading strengths to keep is a separate decision with no universal cutoff. Operational check: rebuild a simplified matrix from the retained orthogonal factors and the retained diagonal strengths, and confirm the factors used in that rebuild are orthogonal.