Question

Which of the following decompositions decomposes any matrix into an orthogonal, a diagonal and an orthogonal matrix? OLU Decomposition Singular Value Decomposition QR Decomposition Eigen Decomposition

Answer 1

1.the singular value decomposition decomposes any matrix into a orthogonal matrix

2.A=$\bigcup \sum$V^T IN THIS FORM THE COLUMN OF V SPANS THE NULL SPACE OF A^T

Theorem 1.5 Let A be an n x d matrix with right singular vectors V1, V2, ..., Vr, left singular vectors u1, U2, ..., Ur, and corresponding singular values 01, 02, ...,Or. Then A= 0;u;v Proof: For each singular vector vj, Av; = {0;u;v} vj. Since any vector v can be ex- pressed as a linear combination of the singular vectors plus a vector perpendicular to the Vi, Av = { 0:0;v[v and by Lemma 1.4, A = Ś0:;v?. The decomposition is called the singular value decomposition, SVD, of A. In matrix notation A=U DVT where the columns of U and V consist of the left and right singular vectors, respectively, and D is a diagonal matrix whose diagonal entries are the singular values of A. For any matrix A, the sequence of singular values is unique and if the singular values are all distinct, then the sequence of singular vectors is unique also. However, when some set of singular values are equal, the corresponding singular vectors span some subspace. Any set of orthonormal vectors spanning this subspace can be used as the singular vectors.