Editing Singular value decomposition (section)

== Example ==
Consider the {{tmath|4 \times 5}} matrix

<math display=block>
\mathbf{M} = \begin{bmatrix}
  1 & 0 & 0 & 0 & 2 \\
  0 & 0 & 3 & 0 & 0 \\
  0 & 0 & 0 & 0 & 0 \\
  0 & 2 & 0 & 0 & 0
\end{bmatrix}
</math>

A singular value decomposition of this matrix is given by {{tmath|\mathbf U\mathbf \Sigma \mathbf V^*}}

<math display=block>\begin{align}
\mathbf{U} &= \begin{bmatrix}
  \color{Green}0 & \color{Blue}-1 & \color{Cyan}0 & \color{Emerald}0 \\
  \color{Green}-1 & \color{Blue}0 & \color{Cyan}0 & \color{Emerald}0 \\
  \color{Green}0 & \color{Blue}0 & \color{Cyan}0 & \color{Emerald}-1 \\
  \color{Green}0 & \color{Blue}0 & \color{Cyan}-1 & \color{Emerald}0
\end{bmatrix} \\[6pt]

\mathbf \Sigma &= \begin{bmatrix}
  3 &       0  &   0 &                    0  & \color{Gray}\mathit{0} \\
  0 & \sqrt{5} &   0 &                    0  & \color{Gray}\mathit{0} \\
  0 &       0  &   2 &                    0  & \color{Gray}\mathit{0} \\
  0 &       0  &   0 & \color{Red}\mathbf{0} & \color{Gray}\mathit{0}
\end{bmatrix} \\[6pt]

 \mathbf{V}^* &= \begin{bmatrix}
   \color{Violet}0        & \color{Violet}0    & \color{Violet}-1  & \color{Violet}0  &\color{Violet}0 \\
   \color{Plum}-\sqrt{0.2}& \color{Plum}0      & \color{Plum}0    & \color{Plum}0    &\color{Plum}-\sqrt{0.8} \\
   \color{Magenta}0       & \color{Magenta}-1  & \color{Magenta}0 & \color{Magenta}0 &\color{Magenta}0 \\
   \color{Orchid}0           & \color{Orchid}0  & \color{Orchid}0  & \color{Orchid}1  &\color{Orchid}0 \\
   \color{Purple} - \sqrt{0.8} & \color{Purple}0 & \color{Purple}0 & \color{Purple}0 & \color{Purple}\sqrt{0.2}
\end{bmatrix}
\end{align}</math>

The scaling matrix {{tmath|\mathbf \Sigma}} is zero outside of the diagonal (grey italics) and one diagonal element is zero (red bold, light blue bold in dark mode). Furthermore, because the matrices {{tmath|\mathbf U}} and {{tmath|\mathbf V^*}} are [[unitary matrix|unitary]], multiplying by their respective conjugate transposes yields [[identity matrix|identity matrices]], as shown below.  In this case, because {{tmath|\mathbf U}} and {{tmath|\mathbf V^*}} are real valued, each is an [[orthogonal matrix]].

<math display=block>\begin{align}
  \mathbf{U} \mathbf{U}^* &=
  \begin{bmatrix}
    1 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0 \\
    0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 1
  \end{bmatrix} = \mathbf{I}_4 \\[6pt]
  \mathbf{V} \mathbf{V}^* &=
  \begin{bmatrix}
    1 & 0 & 0 & 0 & 0 \\
    0 & 1 & 0 & 0 & 0 \\
    0 & 0 & 1 & 0 & 0 \\
    0 & 0 & 0 & 1 & 0 \\
    0 & 0 & 0 & 0 & 1
  \end{bmatrix} = \mathbf{I}_5
\end{align}</math>

This particular singular value decomposition is not unique.  For instance, we can keep {{tmath|\mathbf U}} and {{tmath|\mathbf \Sigma}} the same, but change the last two rows of {{tmath|\mathbf V^*}} such that

<math display=block>\mathbf{V}^* = \begin{bmatrix}
   \color{Violet}0        & \color{Violet}0    & \color{Violet}-1  & \color{Violet}0  &\color{Violet}0 \\
   \color{Plum}-\sqrt{0.2}& \color{Plum}0      & \color{Plum}0    & \color{Plum}0    &\color{Plum}-\sqrt{0.8} \\
   \color{Magenta}0       & \color{Magenta}-1  & \color{Magenta}0 & \color{Magenta}0 &\color{Magenta}0 \\
   \color{Orchid}\sqrt{0.4} &  \color{Orchid}0 & \color{Orchid}0  & \color{Orchid}\sqrt{0.5} & \color{Orchid}-\sqrt{0.1}  \\
  \color{Purple}-\sqrt{0.4} &  \color{Purple}0 & \color{Purple}0  & \color{Purple}\sqrt{0.5} &  \color{Purple}\sqrt{0.1}
\end{bmatrix}</math>

and get an equally valid singular value decomposition. As the matrix {{tmath|\mathbf M}} has rank 3, it has only 3 nonzero singular values. In taking the product {{tmath|\mathbf{U}\mathbf{\Sigma} \mathbf{V}^* }}, the final column of {{tmath|\mathbf U}} and the final two rows of {{tmath|\mathbf{V^*} }} are multiplied by zero, so have no effect on the matrix product, and can be replaced by any unit vectors which are orthogonal to the first three and to each-other.

The [[#Compact SVD|compact SVD]], {{tmath|1= \mathbf M = \mathbf{U}_r\mathbf{\Sigma}_r \mathbf{V}_r^* }}, eliminates these superfluous rows, columns, and singular values:

<math display=block>\begin{align}
\mathbf{U}_r &= \begin{bmatrix}
  \color{Green}0 & \color{Blue}-1 & \color{Cyan}0 \\
  \color{Green}-1 & \color{Blue}0 & \color{Cyan}0 \\
  \color{Green}0 & \color{Blue}0 & \color{Cyan}0 \\
  \color{Green}0 & \color{Blue}0 & \color{Cyan}-1
\end{bmatrix} \\[6pt]

\mathbf \Sigma_r &= \begin{bmatrix}
  3 &       0  &   0 \\
  0 & \sqrt{5} &   0 \\
  0 &       0  &   2
\end{bmatrix} \\[6pt]

 \mathbf{V}^*_r &= \begin{bmatrix}
   \color{Violet}0        & \color{Violet}0    & \color{Violet}-1  & \color{Violet}0  &\color{Violet}0 \\
   \color{Plum}-\sqrt{0.2}& \color{Plum}0      & \color{Plum}0    & \color{Plum}0    &\color{Plum}-\sqrt{0.8} \\
   \color{Magenta}0       & \color{Magenta}-1  & \color{Magenta}0 & \color{Magenta}0 &\color{Magenta}0 
\end{bmatrix}
\end{align}</math>