6 De64257nition A ny r eal matrix can be decomposed uniquely as UDV is and column orthogonal its columns are eigen ve ctors of AA AA UDV VDU UD is and orthogonal its columns are eigen ve ctors of VDU UDV VD is diagonal nonne ga ti ve r eal v alues ca ID: 30370 Download Pdf

6 De64257nition A ny r eal matrix can be decomposed uniquely as UDV is and column orthogonal its columns are eigen ve ctors of AA AA UDV VDU UD is and orthogonal its columns are eigen ve ctors of VDU UDV VD is diagonal nonne ga ti ve r eal v alues ca

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Singular V alue Decomposition (SVD) (T rucco, Appendix A.6) Deﬁnition -A ny r eal matrix can be decomposed uniquely as UDV is and column orthogonal (its columns are eigen ve ctors of AA AA UDV VDU UD is and orthogonal (its columns are eigen ve ctors of VDU UDV VD is diagonal (non-ne ga ti ve r eal v alues called singular va lues) diag ,..., )o rdered so that ... (if is a singular v alue of ,i t ss quare is an eigen va lue of -I ... )a nd ... ), then (actually ,t he sum goes from 1 to where is the rank of An example ,t hen AA 10 10 17 10 10 The eigen va lues of AA are:

28. 86 0. 14 The eigen ve ctors of AA are: 0. 454 0. 766 0. 454 0. 542 0. 643 0. 542 0. 707 0. 707

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-2- The e xpansion of is Important: note that the second eigen va lue is much smaller than the ﬁrst; if we ne glect it from the abo ve s ummation, we can represent by introducing rela- ti ve ly small errors only: 1. 11 1. 87 1. 11 1. 87 3. 15 1. 87 1. 11 1. 87 1. 11 Computing the rank using SVD -T he rank of a matrix is equal to the number of non-zero singular v alues. Computing the in ve rse of a matrix using SVD -As quare matrix is nonsingular if 0f or all -I is a

nonsingular matrix, then its in ve rse is gi ve nb UDV or VD where diag ,..., -I is singular or ill-conditioned, then we can use SVD to approximate its in ve rse by the follo wing matrix: UDV VD 1/ if otherwise (where is a small threshold)

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-3- The condition of a matrix -C onsider the system of linear equations Ax If small changes in can lead to relati ve ly lar ge changes in the solution ,t hen we call Ai ll-conditioned -T he ratio gi ve nb elo wi sr elated to the condition of and measures the de gree of singularity of (the lar ger this v alue is, the closer is to being

singular) (lar gest o ve rs mallest singular v alues) Least Squar es Solutions of Systems -C onsider the ov er -determined system of linear equations Ax ,( is with -L et be the residual v ector for some Ax -T he v ector which yields the smallest possible residual is called a least- squar es solution (it is an approximate solution). || || || Ax || || Ax || for all -A lthough a least-squares solution al wa ys e xist, it might not be unique ! -T he least-squares solution with the smallest norm || || is unique and it is gi ve nb y: Ax or Example: 11

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-4- .1 48 .1 64 .1 80 .1 89 .2

46 .1 07 2. 492 0. 787 Computing using SVD -I is ill-conditioned or singular ,w ec an use SVD to obtain a least squares solution as follo ws: VD 1/ if otherwise (where is a small threshold) Least Squar es Solutions of Systems -I is ill-conditioned or singular ,S VD can gi ve u saw orkable solution in this case too: VD Homogeneous Systems -S uppose =0, then the linear system is called homogeneous: Ax (assume is and UDV -T he minimum-norm solution in this case is =0 (tri vial solution). -F or homogeneous linear systems, the meaning of a least-squares solution is modiﬁed by imposing the

constraint: || || -T his is a "constrained" optimization problem: min || || || Ax ||

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-5- -T he minimum-norm solution for homogeneous systems is not al wa ys unique. Special case: rank 1( 1, =0) solution is av is a constant) is the last column of -- corresponds to the smallest General case: rank ... 0) solution is ... si sac onstant) with ...

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