Schmidt Eckart Young Mirsky Beste Approximatino. Best approximation is xk = u⌃kv⇤ where⌃k is the diagonal matrix with singular values 1, 2,.,k, in the sense that km xkkf =min{km xkf; Where x = u vt is thesingular value decomposition (svd)of x.

Stewart and sun 1990) and it provides an optimal truncation of the propagator in the euclidean norm at any ﬁxed timet. Where x = u vt is thesingular value decomposition (svd)of x. X ˇxb= x r j=1 ˙ ju jv t;

Best Approximation Is Xk = U⌃Kv⇤ Where⌃K Is The Diagonal Matrix With Singular Values 1, 2,.,K, In The Sense That Km Xkkf =Min{Km Xkf;

Stewart (1993) calls it schmidt approximation theorem. X ˇxb= x r j=1 ˙ ju jv t; The method involves computing the reachability and observability gramians.

X 2Mm,N, Rankx = K} (This Theorem Is Due To Eckart And Young, 1936).

Mirsky menggeneralisasi teorema pada tahun 1958 untuk semua norma yang tidak berubah di bawah transformasi kesatuan, dan ini termasuk norma 2 operator. In this paper it is shown how to obtain a best approximation of lower rank in which a specified set of columns of the matrix remains fixed. Since km xkf = ku⌃v⇤ xkf = k⌃ u⇤xvkf denoting n = u⇤xv, an m⇥n matrix of rank k, a direct calculation gives k⌃nk 2 f = x i,j |⌃i,j ni,j| 2 =

The Approximation Error Is X Xb 2 = ˙ R+1:

An alternative solution expression for the generalized matrix approximation problem is obtained. Image compression by truncated svd example: Mirsky, 1958, symmetric gauge functions and unitarily invariant norms

Abstract In Undeterminedin This Paper Theoretical Results Regarding A Generalized Minimum Rank Matrix Approximation Problem In The Spectral Norm Are Presented.

The approximation error is kx xbk 2 = ˙ r+1: Stewart and sun 1990) and it provides an optimal truncation of the propagator in the euclidean norm at any ﬁxed timet. Theorem 1 let kkbe a unitarily invariant norm on m m;n.

The Rank Constraint Is Related To A.

If k<rand a k = p k i=1 ˙ iu iv t i;then it holds: This alternative expression provides a simple characterization of the achievableminimum rank, which is shown. We prove here that we can go forwards and backwards