Mathematical definition of the SVD X=USVT where U is an m x n matrix,S is an n x n diagonal matrix, and Ir is also an n x n matrix. *The columns of U are called the left singular vectors,{ug, and form an orthonormal basis,so that u;'u;=1 for i=j,and u财u=0fori≠j. The rows of Vr are called the right singular vectors,vp,and form an orthonormal basis. *The elements of S are only nonzero on the diagonal,and are called the singular values.Thus,S=diag(s1,.s). Furthermore,sk>0for1≤k≤r,and sk=0for(件l)≤k≤n. xo-Eusv! =1 The term "closest"means that X minimizes the sum of the squares of the difference of the elements of X and X,Mathematical definition of the SVD where U is an m x n matrix, S is an n x n diagonal matrix, and VT is also an n x n matrix. *The columns of U are called the left singular vectors, {u k}, and form an orthonormal basis, so that ui·uj = 1 for i = j, and ui·uj = 0 for i ≠ j. The rows of VT are called the right singular vectors, {v k}, and form an orthonormal basis. *The elements of S are only nonzero on the diagonal, and are called the singular values. Thus, S = diag(s1 ,...,s n). Furthermore, sk > 0 for 1 ≤ k ≤ r, and sk = 0 for ( r+1) ≤ k ≤ n. The term “closest” means that X (l) minimizes the sum of the squares of the difference of the elements of X and X(l)