The Extraordinary SVD

: We review the singular value decomposition (SVD) and discuss some lesser-known applications of it that we find particularly interesting. We also discuss generalizations of the SVD and hope to convey through these generalizations and the selected examples why the SVD is so extraordinary. Let’s start with one of our favorite theorems from linear algebra and what is perhaps the most important theorem in this paper.1 Theorem 1 Any matrix A ∈ Rm×n can be factored into a singular value decomposition (SVD), A = USV T , (1) where U ∈ Rm×m and V ∈ Rn×n are orthogonal matrices (i.e., UUT = V V T = I) and S ∈ Rm×n is diagonal with r = rank(A) leading positive diagonal entries. The diagonal entries of S are called the singular values of A. The nonzero singular values of A are the square roots of the nonzero eigenvalues of both AAT and AT A. ∗ Carla D. Martin is an Assistant Professor in the Department of Mathematics and Statistics at James Madison University. † Mason Porter is...

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Carla D. Martin, Mason A. Porter
• Keywords: CORR 2011, Education, James Madison University, Singular Value Decomposition, Singular Value Decomposition Svd