Editing Singular value decomposition (section)

=== Geometric meaning ===
Because {{tmath|\mathbf U}} and {{tmath|\mathbf V}} are unitary, we know that the columns {{tmath|\mathbf U_1, \ldots, \mathbf U_m}} of {{tmath|\mathbf U}} yield an [[orthonormal basis]] of {{tmath|K^m}} and the columns {{tmath|\mathbf V_1, \ldots, \mathbf V_n}} of {{tmath|\mathbf V}} yield an orthonormal basis of {{tmath|K^n}} (with respect to the standard [[scalar product]]s on these spaces).

The [[linear transformation]]

<math display=block>
T : \left\{\begin{aligned}
  K^n &\to K^m \\
  x &\mapsto \mathbf{M}x \end{aligned}\right.
</math>

has a particularly simple description with respect to these orthonormal bases: we have

<math display=block>
T(\mathbf{V}_i) = \sigma_i \mathbf{U}_i, \qquad i
= 1, \ldots, \min(m, n),
</math>

where {{tmath|\sigma_i}} is the {{tmath|i}}-th diagonal entry of {{tmath|\mathbf \Sigma,}} and {{tmath|T(\mathbf V_i) {{=}} 0}} for {{tmath|i > \min(m,n).}}

The geometric content of the SVD theorem can thus be summarized as follows: for every linear map {{tmath|T : K^n \to K^m }} one can find orthonormal bases of {{tmath|K^n}} and {{tmath|K^m}} such that {{tmath|T}} maps the {{tmath|i}}-th basis vector of {{tmath|K^n}} to a non-negative multiple of the {{tmath|i}}-th basis vector of {{tmath|K^m,}} and sends the leftover basis vectors to zero. With respect to these bases, the map {{tmath|T}} is therefore represented by a diagonal matrix with non-negative real diagonal entries.

To get a more visual flavor of singular values and SVD factorization – at least when working on real vector spaces – consider the sphere {{tmath|S}} of radius one in {{tmath|\mathbf R^n.}} The linear map {{tmath|T}} maps this sphere onto an [[ellipsoid]] in {{tmath|\mathbf R^m.}} Non-zero singular values are simply the lengths of the [[Semi-minor axis|semi-axes]] of this ellipsoid. Especially when {{tmath|n {{=}} m,}} and all the singular values are distinct and non-zero, the SVD of the linear map {{tmath|T}} can be easily analyzed as a succession of three consecutive moves: consider the ellipsoid {{tmath|T(S)}} and specifically its axes; then consider the directions in {{tmath|\mathbf R^n}} sent by {{tmath|T}} onto these axes. These directions happen to be mutually orthogonal. Apply first an isometry {{tmath|\mathbf V^*}} sending these directions to the coordinate axes of {{tmath|\mathbf R^n.}} On a second move, apply an [[endomorphism]] {{tmath|\mathbf D}} diagonalized along the coordinate axes and stretching or shrinking in each direction, using the semi-axes lengths of {{tmath|T(S)}} as stretching coefficients. The composition {{tmath|\mathbf D \circ \mathbf V^*}} then sends the unit-sphere onto an ellipsoid isometric to {{tmath|T(S).}} To define the third and last move, apply an isometry {{tmath|\mathbf U}} to this ellipsoid to obtain {{tmath|T(S).}} As can be easily checked, the composition {{tmath|\mathbf U \circ \mathbf D \circ \mathbf V^*}} coincides with {{tmath|T.}}