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Unformatted text preview: 7  1 Singular Values and Matrix Norms S. Lall, Stanford 2009.10.07.02 7. Singular Values and Matrix Norms Geometry of linear maps The singular value decomposition (SVD) Interpretation of the SVD Matrix properties via the SVD, rank, range and null space Example: testing achievability of desired outputs for control The SVD and the control ellipsoid Example: forces applied to a hovercraft Summary: svd, control and estimation The matrix norm, and inequalities Example: matrix norm and estimation Minimalrank approximation Example: minimalrank approximation Example: image compression 7  2 Singular Values and Matrix Norms S. Lall, Stanford 2009.10.07.02 The key points of this section there is an ellipsoid for control problems, which tells which directions have strong and weak actuator authority ellipsoids in both control and estimation problems arise because of the geometry of linear transformations the singular value decomposition gives a way to both compute and understand this geometry the svd also gives us a way to pick out the essential features of any linear map, and simplify it by remove the unessentials 7  3 Singular Values and Matrix Norms S. Lall, Stanford 2009.10.07.02 Geometry of Linear Maps v 1 v 2 1 u 1 2 u 2 ! an extremely important fact: every matrix A R m n maps the unit ball in R n to an ellipsoid in R m S = braceleftBig x R n vextendsingle vextendsingle bardbl x bardbl 1 bracerightBig AS = braceleftBig Ax vextendsingle vextendsingle x S bracerightBig 7  4 Singular Values and Matrix Norms S. Lall, Stanford 2009.10.07.02 singular values and singular vectors first, assume A R m n is skinny and full rank v 1 v 2 1 u 1 2 u 2 ! the numbers 1 ,..., n are called the singular values of A by convention, i > the vectors u 1 ,...,u n are called the left singular vectors of A these are unit vectors along the principal semiaxes of AS the vectors v 1 ,...,v n are called the right singular vectors of A these are the preimages of the principal semiaxes, defined so that Av i = i u i 7  5 Singular Values and Matrix Norms S. Lall, Stanford 2009.10.07.02 The Thin Singular Value Decomposition we have A R m n , skinny and full rank, and Av i = i u i for 1 i n let U = bracketleftbig u 1 u 2 u n bracketrightbig = 1 2 . . . n V = bracketleftbig v 1 v 2 v n bracketrightbig in matrix form, the above equation is AV = U and since V is orthogonal A = U V T called the thin (or reduced) SVD of A V T = A U 7  6 Singular Values and Matrix Norms S. Lall, Stanford 2009.10.07.02 The Full Singular Value Decomposition we can add extra orthonormal columns to U to make U = bracketleftbig u 1 u 2 u m bracketrightbig an orthogonal matrix; we also add extra rows of zeros to , so A = U V T V T = A U this is the (full) singular value decomposition...
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