Posting this here as reference material in regard to my present issues with the
matrix V-transpose.

SVD calculator used is:
http://engineerjs.com/?sidebar=doc/ejs/engine/linalg-1/_svd.html

The data and script can be copied and pasted:
------------------------------------------------------------------------------

A = [0.09, -0.41, 0.42, 0.43, -0.40, 0.18, -0.01, 0.35, 0.19, 0.27, -0.43, 0.49,
-0.17, -0.35, -0.06, -0.01, -0.35, 0.50, -0.23, 0.31, -0.49, 0.43, -0.32, -0.40,
 0.24, 0.07, -0.43, 0.35, 0.51, -0.38, 0.40, 0.30, 0.39, 0.23, -0.22, 0.34,
-0.38, -0.43, -0.14, 0.29, -0.24, 0.21, 0.25, -0.28, -0.37, -0.14, -0.42, -0.27,
 -0.39, 0.42, 0.42, 0.01, -0.04, 0.02, -0.17, -0.43, -0.41, 0.22, -0.31, -0.27,
-0.44, 0.33, -0.51, -0.03, -0.46, 0.32, -0.32, 0.34, 0.21, 0.47, -0.37, -0.42,
 0.19, 0.05, -0.02, 0.45, 0.10, 0.42, -0.25, 0.24, -0.13, -0.36, 0.50, -0.20,
-0.20, -0.35, 0.34, 0.42, -0.47, -0.41, 0.37, 0.09, 0.46, -0.41, 0.41, 0.35,
 -0.24, -0.11, -0.10, 0.26, -0.02, -0.18, -0.33, 0.48, 0.24, -0.02, -0.14,
-0.23, -0.51, -0.12, 0.17, -0.39, 0.25, -0.50, 0.26, 0.06, 0.44, -0.02, -0.47,
0.01,
 0.41, 0.40, 0.36, -0.40, -0.27, -0.46, 0.06, 0.47, -0.16, -0.33, 0.14, -0.38,
0.24, 0.06, 0.38, 0.50, 0.01, 0.42, -0.24, 0.23, 0.24, -0.28, -0.48, 0.25, 0.49,
 -0.18, 0.36, -0.15, -0.21, 0.06, -0.17, 0.50, 0.31, 0.06, 0.01, 0.27, -0.31,
-0.01, 0.26, -0.51, -0.27, 0.22, -0.26, -0.06, 0.33, -0.23, -0.42, 0.33, -0.03,
 0.39, 0.41, 0.26, -0.17, 0.25, -0.40, -0.12, -0.07, -0.20, -0.42, -0.46, -0.10,
-0.00, 0.51, -0.14, -0.41, 0.24, -0.34, 0.36, -0.28, -0.14, -0.48, 0.09, 0.28,
 0.38, -0.28, -0.08, 0.48, -0.06, 0.06, 0.03, 0.40, 0.31, 0.09, -0.08, 0.36,
-0.45, -0.24, 0.36, 0.28, 0.19, 0.05, -0.50, -0.29, 0.48, -0.09, 0.35, 0.12,
-0.41,
 0.35, 0.50, -0.33, -0.22, 0.43, -0.24, -0.27, 0.44, -0.45, 0.12, 0.32, -0.40,
0.13, -0.08, -0.48, 0.08, 0.46, 0.14, -0.18, -0.23, -0.22, -0.10, 0.27, 0.40,
 -0.16, 0.03, -0.38, -0.46, 0.37, 0.06, 0.38, -0.24, -0.23, -0.29, 0.02, -0.26,
0.38, -0.08, 0.38, -0.33, 0.08, -0.11, -0.37, 0.15, -0.13, 0.50, 0.40, 0.47,
 0.36, -0.40, -0.17, -0.30, -0.23, -0.42, 0.06, 0.33, -0.47, 0.17, 0.34, -0.11,
-0.22, 0.44, -0.02, 0.25, 0.05, -0.29, -0.49, -0.43, 0.03, 0.36, -0.32, 0.47,
 -0.22, -0.20, -0.26, 0.49, 0.25, 0.28, 0.24, -0.21, 0.14, -0.13, -0.20, -0.30,
-0.17, -0.38, 0.32, 0.17, 0.40, -0.10, -0.38, -0.16, 0.36, 0.46, -0.27, 0.09,
 -0.10, -0.41, 0.47, 0.29, 0.24, 0.25, 0.50, -0.27, -0.16, -0.09, -0.41, 0.20,
0.41, -0.33, -0.38, 0.12, -0.18, 0.08, 0.12, 0.41, -0.33, 0.41, -0.20, -0.13,
 -0.51, -0.43, -0.32, 0.22, 0.16, 0.36, -0.15, 0.04, -0.09, 0.25, -0.19, -0.27,
-0.19, 0.05, 0.38, 0.03, 0.06, 0.38, -0.41, -0.27, -0.33, -0.05, 0.49, 0.12,
 0.22, 0.24, 0.25, -0.05, 0.24, -0.42, 0.34, -0.05, -0.49, -0.46, -0.11, 0.34,
0.38, -0.10, -0.42, 0.19, -0.07, -0.18, -0.41, 0.46, -0.33, -0.48, 0.01, 0.03,
 0.26, -0.17, -0.46, 0.08, 0.32, 0.34, -0.03, 0.19, 0.15, -0.39, -0.18, -0.04,
-0.12, -0.20;
-0.33, 0.10, 0.02, 0.12, -0.43, -0.25, 0.45, 0.73, -0.17, 0.90, -0.65, 0.41,
0.05, 0.18, -0.37, 0.53, 0.13, 0.35, -0.57, -0.22, -0.47, 0.80, 0.08, 0.12,
-0.25,
 -0.34, -0.15, 0.22, 0.34, -0.67, -0.03, 0.78, 0.46, -0.10, -0.82, -0.18, -0.74,
-0.82, 0.59, 0.57, 0.14, 0.83, 0.53, -0.41, -0.71, 0.17, -0.51, -0.91, -0.11,
 0.75, 0.40, 0.04, -0.78, 0.59, -0.90, -0.25, -0.44, -0.24, -0.51, 0.29, -0.69,
-0.27, -0.56, 0.32, -0.28, 0.46, -0.41, 0.02, 0.74, 0.65, -0.81, -0.46, 0.21,
 -0.74, 0.76, 0.48, 0.58, 0.24, 0.10, 0.90, 0.47, -0.88, 0.53, -0.91, -0.91,
0.10, -0.14, 0.21, -0.31, -0.00, 0.71, 0.33, 0.48, 0.02, -0.08, 0.37, 0.47,
0.29,
 0.07, -0.42, -0.06, -0.85, -0.20, 0.45, 0.47, 0.76, 0.58, -0.24, -0.40, 0.65,
-0.45, 0.02, 0.78, -0.35, -0.23, 0.14, 0.05, 0.26, -0.74, 0.38, 0.55, -0.13,
 0.47, 0.11, -0.44, -0.10, -0.54, 0.20, 0.13, -0.91, -0.32, -0.40, 0.90, 0.82,
-0.04, 0.38, 0.47, 0.83, 0.47, 0.03, -0.24, 0.04, -0.21, 0.73, 0.44, -0.89,
 -0.11, -0.34, 0.04, -0.29, -0.60, 0.62, 0.38, -0.66, -0.48, 0.09, -0.36, 0.50,
-0.43, -0.40, -0.83, -0.51, 0.27, 0.25, -0.30, -0.30, -0.83, -0.26, 0.38, 0.48,
 0.05, 0.55, 0.57, 0.01, -0.33, 0.57, -0.60, -0.12, -0.80, -0.16, 0.31, 0.10,
0.52, -0.19, -0.71, 0.17, -0.74, -0.12, 0.29, -0.76, -0.26, -0.34, 0.81, 0.72,
 -0.86, 0.12, 0.72, -0.83, -0.08, 0.58, 0.86, 0.17, -0.32, 0.34, 0.05, -0.68,
0.44, 0.88, 0.82, 0.03, -0.08, -0.47, -0.43, 0.25, 0.69, 0.56, -0.65, -0.14,
0.10,
 0.34, -0.86, 0.49, 0.50, 0.04, -0.92, 0.67, -0.07, -0.01, 0.91, 0.10, 0.85,
0.70, -0.52, 0.64, 0.62, 0.26, 0.14, 0.14, 0.50, 0.66, 0.88, 0.75, -0.39, -0.57,
 -0.60, -0.19, -0.17, 0.82, 0.30, -0.88, -0.79, -0.83, -0.76, -0.26, 0.88,
-0.27, 0.06, -0.59, 0.19, -0.52, -0.63, 0.83, 0.39, 0.28, -0.09, 0.44, -0.05,
0.09,
 -0.88, -0.09, 0.45, -0.18, 0.65, -0.16, -0.66, 0.13, -0.03, 0.49, -0.48, 0.08,
0.76, 0.69, 0.82, -0.83, -0.64, -0.21, -0.74, 0.12, -0.67, 0.64, -0.90, -0.45,
 -0.08, 0.22, 0.72, -0.22, 0.90, 0.17, -0.59, 0.18, 0.54, -0.08, 0.50, -0.62,
0.90, -0.53, -0.11, 0.32, -0.30, 0.40, 0.01, 0.11, 0.13, -0.21, 0.44, -0.70,
0.72,
 -0.08, 0.35, 0.47, 0.30, -0.64, 0.61, 0.63, -0.75, 0.27, 0.72, -0.63, 0.14,
0.67, -0.48, 0.84, 0.80, 0.66, -0.71, 0.74, 0.43, -0.35, -0.50, -0.30, -0.52,
0.78,
 -0.58, 0.18, -0.52, -0.75, -0.19, -0.42, -0.42, -0.88, 0.54, 0.25, 0.75, 0.67,
-0.61, 0.05, 0.08, -0.91, 0.26, 0.49, 0.24, -0.10, 0.80, 0.85, -0.33, 0.69,
 -0.16, -0.02, 0.33, -0.83, -0.62, -0.29, -0.38, 0.87, 0.32, 0.68, 0.05, 0.86,
0.47, -0.67, -0.26, 0.07, 0.16, -0.71, -0.74, 0.01, 0.91, -0.54, -0.52, -0.42,
 -0.30, 0.46, -0.24, 0.13, -0.54, -0.83, -0.08, 0.10, -0.70, 0.52;
-0.34, 0.24, -0.10, -0.08, 0.00, -0.36, -0.32, -0.04, 0.22, -0.11, 0.10, -0.38,
-0.25, 0.04, -0.29, 0.24, 0.01, -0.22, -0.13, -0.32, 0.16, -0.34, -0.06, 0.23,
 -0.35, 0.30, 0.07, -0.46, -0.28, 0.04, -0.08, -0.02, 0.01, 0.19, 0.33, -0.02,
0.41, 0.24, -0.06, 0.08, -0.16, -0.32, -0.45, 0.47, 0.43, 0.39, 0.05, 0.24,
0.42,
 -0.40, -0.46, -0.38, 0.19, -0.30, 0.13, 0.43, 0.03, 0.16, -0.08, -0.06, 0.14,
-0.08, 0.33, -0.34, 0.10, 0.09, -0.09, 0.07, 0.07, -0.19, 0.38, 0.46, -0.46,
 0.01, -0.01, -0.44, 0.20, -0.03, 0.41, -0.11, -0.19, 0.20, -0.38, 0.16, 0.19,
-0.02, -0.34, -0.03, 0.13, 0.07, -0.05, -0.41, -0.43, 0.34, -0.18, 0.07, 0.07,
 -0.27, -0.30, -0.18, -0.36, 0.01, -0.09, -0.16, -0.46, 0.00, -0.10, -0.19,
0.32, -0.02, 0.15, 0.37, 0.02, 0.20, -0.35, -0.41, -0.14, 0.34, 0.32, -0.35,
-0.03,
 -0.21, -0.47, 0.26, -0.14, 0.21, 0.24, -0.16, -0.25, 0.23, 0.23, 0.47, -0.26,
-0.17, -0.37, -0.23, -0.33, -0.30, 0.05, 0.20, -0.36, 0.42, 0.19, -0.40, -0.18,
 0.23, -0.03, -0.24, -0.22, -0.34, 0.41, -0.29, 0.10, -0.13, 0.31, 0.16, 0.48,
0.26, -0.13, 0.33, 0.03, -0.10, 0.33, -0.32, -0.09, 0.47, 0.25, -0.04, -0.32,
 -0.47, -0.38, -0.45, 0.13, -0.44, -0.01, -0.13, 0.34, 0.45, 0.19, 0.15, 0.34,
-0.37, -0.26, 0.44, 0.09, -0.46, 0.03, -0.34, 0.30, 0.32, 0.36, -0.33, -0.36,
 -0.42, 0.33, -0.32, -0.30, 0.03, -0.39, 0.21, -0.24, -0.46, -0.34, 0.33, 0.04,
0.40, 0.20, -0.18, -0.03, 0.24, -0.38, 0.19, -0.12, -0.35, -0.04, 0.03, 0.04,
 0.03, 0.05, -0.34, 0.34, 0.16, -0.45, 0.43, 0.17, -0.42, 0.32, -0.42, -0.26,
0.27, -0.02, -0.04, 0.41, -0.33, -0.14, 0.26, -0.23, -0.17, 0.12, 0.09, -0.31,
 -0.11, -0.23, -0.22, 0.02, 0.37, -0.10, -0.17, 0.03, 0.07, -0.03, 0.05, -0.00,
0.47, -0.25, -0.30, -0.41, -0.04, -0.41, -0.22, 0.01, -0.27, 0.31, -0.28, -0.28,
 -0.13, 0.01, 0.28, 0.22, 0.45, -0.02, 0.42, -0.31, 0.02, 0.18, 0.25, -0.41,
-0.20, -0.16, -0.12, 0.02, 0.07, -0.02, 0.36, 0.23, 0.42, 0.11, 0.04, -0.02,
-0.16,
 0.10, -0.19, 0.45, -0.31, 0.05, -0.34, -0.27, 0.39, -0.14, 0.39, 0.09, 0.45,
0.22, 0.02, -0.17, -0.19, -0.17, 0.34, 0.47, 0.30, 0.03, -0.31, -0.12, -0.37,
 -0.24, 0.09, -0.34, -0.41, -0.47, -0.46, -0.21, -0.07, 0.07, -0.12, 0.40,
-0.46, -0.41, -0.03, 0.08, -0.35, 0.44, -0.14, 0.05, -0.43, 0.01, -0.12, 0.26,
0.43,
 0.28, 0.44, 0.47, -0.36, -0.14, 0.06, 0.41, 0.03, 0.42, -0.21, 0.45, 0.08,
0.15, 0.31, -0.08, -0.27, 0.20, -0.00, 0.17, 0.24, 0.25, 0.27, -0.33, 0.29,
-0.34,
 -0.32, -0.30, 0.12, -0.39, 0.11, 0.08, 0.03, 0.38, 0.41, 0.42, -0.32, -0.47,
0.01, 0.27, -0.03, -0.26, 0.39, 0.45, -0.20, -0.02, 0.35, 0.15, 0.35, -0.25,
 -0.18, 0.43, 0.26, -0.20, -0.47, 0.39, -0.45, -0.20, 0.16, -0.24, 0.38, -0.13,
0.00
];

  print("Non-square matrix").title()
  A
  print("Calculate S, U, and V-transpose").title()
  s = ejs.svd(A,true,true)

  print("Show that U-transpose x U = I (small numbers = 0)").title()
  s.u'*s.u //!!

  print("Show that V x V-transpose = I (small numbers = 0)").title()
  s.v*s.v'

  print("Show that U x S x V = A, the original matrix").title()
  s.u*diag(s.d)*s.v

------------------------------------------------------------------------------

Running that will give you U, S, and V (presumably V-transpose, but it hardly
matters, as the matrix has scaling and is not purely rotational)

The first three columns get copied into POV-Ray for use as a transform matrix as
follows:

#declare VC1 = <-0.04816, -0.07305, -0.01401>;
#declare VC2 = <-0.03091,  0.07824, -0.01210>;
#declare VC3 = < 0.06011, -0.05455,  0.01822>;

#declare VR1 = <VC1.x, VC2.x, VC3.x>;
#declare VR2 = <VC1.y, VC2.y, VC3.y>;
#declare VR3 = <VC1.z, VC2.z, VC3.z>;

#declare V = transform {
matrix
<
 VR1.x, VR2.x, VR3.x, //
 VR1.y, VR2.y, VR3.y, //
 VR1.z, VR2.z, VR3.z, //
 0.000, 0.000, 0.000  //
>
}

Attached is a render showing - in the strip between -1 and 1 from top to bottom
- a sphere transformed by U.  No change, as it's a purely rotational matrix.

Two small ellipsoids.   These are created by taking a sphere and transforming
with V and V-transpose.   Obvious scaling takes place.

One larger ellipsoid - created by applying V and then S.

Applying only S gives an ellipsoid that is so large that it blots out everything
else in the render.

Post a reply to this message

Attachments:
Download 'svd_ellipse_test.png' (316 KB)

Preview of image 'svd_ellipse_test.png'