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CS 395/495-26: Spring 2004 ,[object Object],[object Object],[object Object]
Matrix Multiply: A Change-of-Axes ,[object Object],[object Object],[object Object],x 1 … x n a 11 … a 1n … … a m1 … a mn = b 1 … b m x 1 x 2 b 3 b 2 b 1 Ax=b x b Input (N dim) (2D) Output (M-dim) (3D)
Matrix Multiply: A Change-of-Axes ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],x 1 … x n a 11 … a 1n … … a m1 … a mn = b 1 … b m x 1 x 2 A 2 A 1 A 3 b 3 b 2 b 1 Ax=b x b Input (N dim) (2D) Output (M-dim) (3D)
[object Object],[object Object],[object Object],How Does Matrix ‘stretch space’? x 1 x 2 b 3 b 2 b 1 Ax=b 1 Input (N dim.) Output (M dim.) u 1 u 2 x 1 ellipsoid (disk) b
[object Object],[object Object],[object Object],[object Object],SVD finds: ‘Output’ Vectors U , and… x 1 x 2 b 3 b 2 b 1 Ax=b 1 Input (N dim.) Output (M dim.) u 1 u 2 x 1 ellipsoid (disk) b
SVD finds: …‘Input’ Vectors V , and… ,[object Object],[object Object],[object Object],[object Object],x 1 x 2 Ax=b x Input (N dim.) b 3 b 2 b 1 Output (M dim.) u 1 u 2 v 2 v 1 b
SVD finds: …‘Scale’ factors S . ,[object Object],[object Object],[object Object],[object Object],x 1 x 2 Ax=b x Input (N dim.) b 3 b 2 b 1 Output (M dim.) u 1 u 2 v 2 v 1 s 2 s 1 b
SVD Review: What is it? ,[object Object],[object Object],[object Object],[object Object],x 1 x 2 Ax=b x Input (N dim.) b 3 b 2 b 1 Output (M dim.) u 1 u 3 u 2 v 2 v 1 A = USV T ‘Find Input & Output Axes, Linked by Scale’ b
‘ Let SVDs Explain it All for You’ ,[object Object],[object Object],[object Object],[object Object],x 1 x 2 Ax=b x Input (N dim.) b 3 b 2 b 1 Output (M dim.) u 1 u 3 u 2 v 2 v 1 A = USV T ‘Find Input & Output Axes, Linked by Scale’
‘ Solve for the Null Space’ A x =0 ,[object Object],[object Object],[object Object],[object Object],x 1 x 2 Ax=b x Input (N dim.) b 3 b 2 b 1 Output (M dim.) u 1 u 3 u 2 v 2 v 1 A = USV T ‘Find Input & Output Axes, Linked by Scale’
More on SVDs: Handout ,[object Object],[object Object],[object Object],[object Object]
END

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