svd_2009_10_07_02

# svd_2009_10_07_02 - 7 1 Singular Values and Matrix Norms S...

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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 • Minimal-rank approximation • Example: minimal-rank 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 û 1 u 1 û 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 û 1 u 1 û 2 u 2 ! • the numbers σ 1 ,...,σ n are called the singular values of A by convention, σ 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 = σ 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 = σ i u i for 1 ≤ i ≤ n let ˆ U = bracketleftbig u 1 u 2 ··· u n bracketrightbig ˆ Σ = σ 1 σ 2 . . . σ n V = bracketleftbig v 1 v 2 ··· v n bracketrightbig in matrix form, the above equation is AV = ˆ U ˆ Σ and since V is orthogonal A = ˆ U ˆ Σ V T called the thin (or reduced) SVD of A V T = A U ê Î ê 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 ˆ U to make U = bracketleftbig u 1 u 2 ··· u m bracketrightbig an orthogonal matrix; we also add extra rows of zeros to ˆ Σ , so A = U Σ V T V T = A U Î this is the (full) singular value decomposition...
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svd_2009_10_07_02 - 7 1 Singular Values and Matrix Norms S...

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