CLEVE'S CORNER Professor SVD BY CLEVE MOLER Stanford computer science professor Gene Golub has done more than anyone to make the singular value decomposition one of the most powerful and widely used tools in modern matrix computation from its SVD.Takeo,.,,0/ and=/4.Let APR CA L IF O R NI A 120951 -(m8) PROF SVD -(08) () The matrices Uand Vare roations through anglesandfollowed by reflections in the first dimension.The matrixis a diagonal scaling transformation.Generate A by computing The SVD is a recer lopment.Pete nothing to do with singular matrices A-UEVT rt,auth 3 paper Early Hi the You will find that me that the term the and 196 t used by d1910: d the SVD as a no 1.4015-1.0480 Picard used the way of characterizing A-(40910133） dition number of a matrix.We did not ye or ou the time.it had have a practical way to actually compute This says that the matrix A can be gener- it.Gene Golub and W.Kah- ated by a rotation through 45and a re- an published the first effec flection,followed by independent scalings in each of the two coordinate directi tive algorithm in 1965.A The singular value by factors of 2 and 1/2,respectively,fol- variant of that algorithm. published by Gene Golub lowed by a rotation through3and an- decomposition (SVD) other reflection nd Ch the one we use The MATLAB function generates is a matrix factorization By the time the firs a hgure that demonstrates the singular valu decomposition of a 2-by-2 matrix.Enter the statements with a wide range of We can a 2-bv-2 A·[1.4015-1.0480 interesting applications -0,40091,01331 wards,computing a matrix eigshow(A) Reprinted from TheMathWorks News&Notes Oclober 200wwmathworks.com Reprinted from T heMathWorksNews&Notes | October 2006 | www.mathworks.com Cleve’s Corner By Cleve Moler from its SVD. Take σ1 = 2, σ2 = 1/2, θ = π/6 and φ = π/4. Let U = ( -cos θ sin θ sin θ cos θ) ∑ = ( σ1 0 0 σ2 ) V = ( -cos φ sin φ sin φ cos φ) The matrices U and V are rotations through angles θ and φ, followed by reflections in the first dimension. The matrix ∑ is a diagonal scaling transformation. Generate A by computing A = U∑V T You will find that A = ( 1.4015 -1.0480 - .4009 1.0133 ) This says that the matrix A can be gener￾ated by a rotation through 45° and a re￾flection, followed by independent scalings in each of the two coordinate directions by factors of 2 and 1/2, respectively, fol￾lowed by a rotation through 30° and an￾other reflection. The MATLAB function eigshow generates a figure that demonstrates the singular value decomposition of a 2-by-2 matrix. Enter the statements A = [1.4015 -1.0480; -0.4009 1.0133] eigshow(A) Professor SVD Stanford computer science professor Gene Golub has done more than anyone to make the singular value decomposition one of the most powerful and widely used tools in modern matrix computation. The SVD is a recent development. Pete Stewart, author of the 1993 paper “On the Early History of the Singular Value Decomposition”, tells me that the term valeurs singulières was first used by Emile Picard around 1910 in connection with in￾tegral equations. Picard used the adjective “singular” to mean something exceptional or out of the ordinary. At the time, it had nothing to do with singular matrices. When I was a graduate student in the ear￾ly 1960s, the SVD was still regarded as a fairly obscure theoretical concept. A book that George Forsythe and I wrote in 1964 described the SVD as a nonconstructive way of characterizing the norm and con￾dition number of a matrix. We did not yet have a practical way to actually compute it. Gene Golub and W. Kah￾an published the first effec￾tive algorithm in 1965. A variant of that algorithm, published by Gene Golub and Christian Reinsch in 1970 is still the one we use today. By the time the first MATLAB appeared, around 1980, the SVD was one of its highlights. We can generate a 2-by-2 example by working back￾wards, computing a matrix The singular value decom­position (SVD), is a matrix factorization with a wide range of interesting applications. Gene Golub’s license plate, photographed by Professor P. M. Kroonenberg of Leiden University