Fal!2001 16.3121-3 Singular value decomposition Must perform the SVD of the matrix G(s)at each frequency s=ju G(ju)∈CpmU∈C∑∈RmV∈Cnxm G=U∑Vh UHU=IUUH=I VHV=I VVH=I and 2 is diagonal Diagonal elements ok 20 of 2 are the singular values of G 入(GG)oro i(GGH the positive ones are the same from both formulas The columns of the matrices U and V(; and v;)are the asso- ciated eigenvectors G Gu GGHa o2u 0U If the rank(G)=r< min(p, m), then k>0,k=1,,r Ok=0,k=r+1,., min(p, m An SVD gives a very detailed description of how a ma trix(the system G) acts on a vector (the input w) at a particular frequencyFall 2001 16.31 21—3 Singular Value Decomposition • Must perform the SVD of the matrix G(s) at each frequency s = jω G(jω) ∈ Cp×m U ∈ Cp×p Σ ∈ Rp×m V ∈ Cm×m G = UΣV H — U HU = I, UU H = I, V HV = I, V V H = I, and Σ is diagonal. — Diagonal elements σk ≥ 0 of Σ are the singular values of G. σi = q λi(GHG) or σi = q λi(GGH) the positive ones are the same from both formulas. — The columns of the matrices U and V (ui and vj) are the asso￾ciated eigenvectors GHGvj = σj 2 vj GGHui = σi 2 ui Gvi = σiui • If the rank(G) = r ≤ min(p, m), then — σk > 0, k = 1,..., r — σk = 0, k = r + 1,..., min(p, m) • An SVD gives a very detailed description of how a ma￾trix (the system G) acts on a vector (the input w) at a particular frequency