# Singular Value Decomposition - Example

Example

Consider the 4×5 matrix

$M = begin{bmatrix} 1 & 0 & 0 & 0 & 2\ 0 & 0 & 3 & 0 & 0\ 0 & 0 & 0 & 0 & 0\ 0 & 4 & 0 & 0 & 0end{bmatrix}.$

A singular value decomposition of this matrix is given by

$U = begin{bmatrix} 0 & 0 & 1 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 0 & 1\ 1 & 0 & 0 & 0end{bmatrix} ,; Sigma = begin{bmatrix} 4 & 0 & 0 & 0 & 0\ 0 & 3 & 0 & 0 & 0\ 0 & 0 & sqrt{5} & 0 & 0\ 0 & 0 & 0 & 0 & 0end{bmatrix} ,; V^* = begin{bmatrix} 0 & 1 & 0 & 0 & 0\ 0 & 0 & 1 & 0 & 0\ sqrt{0.2} & 0 & 0 & 0 & sqrt{0.8}\ 0 & 0 & 0 & 1 & 0\ sqrt{0.8} & 0 & 0 & 0 & -sqrt{0.2}end{bmatrix}.$

Notice is zero outside of the diagonal and one diagonal element is zero. Furthermore, because the matrices and are unitary, multiplying by their respective conjugate transposes yields identity matrices, as shown below. In this case, because and are real valued, they each are an orthogonal matrix.

$U U^* = begin{bmatrix} 0 & 0 & 1 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 0 & 1\ 1 & 0 & 0 & 0end{bmatrix} cdot begin{bmatrix} 0 & 0 & 0 & 1\ 0 & 1 & 0 & 0\ 1 & 0 & 0 & 0\ 0 & 0 & 1 & 0end{bmatrix} = begin{bmatrix} 1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1end{bmatrix} equiv I_4$

and

$V V^* = begin{bmatrix} 0 & 0 & sqrt{0.2} & 0 & sqrt{0.8}\ 1 & 0 & 0 & 0 & 0\ 0 & 1 & 0 & 0 & 0\ 0 & 0 & 0 & 1 & 0\ 0 & 0 & sqrt{0.8} & 0 & -sqrt{0.2} end{bmatrix} cdot begin{bmatrix} 0 & 1 & 0 & 0 & 0\ 0 & 0 & 1 & 0 & 0\ sqrt{0.2} & 0 & 0 & 0 & sqrt{0.8}\ 0 & 0 & 0 & 1 & 0\ sqrt{0.8} & 0 & 0 & 0 & -sqrt{0.2}end{bmatrix} = begin{bmatrix} 1 & 0 & 0 & 0 & 0\ 0 & 1 & 0 & 0 & 0\ 0 & 0 & 1 & 0 & 0\ 0 & 0 & 0 & 1 & 0\ 0 & 0 & 0 & 0 & 1end{bmatrix} equiv I_5.$

This particular singular value decomposition is not unique. Choosing such that

$V^* = begin{bmatrix} 0 & 1 & 0 & 0 & 0\ 0 & 0 & 1 & 0 & 0\ sqrt{0.2} & 0 & 0 & 0 & sqrt{0.8}\ sqrt{0.4} & 0 & 0 & sqrt{0.5} & -sqrt{0.1}\ -sqrt{0.4} & 0 & 0 & sqrt{0.5} & sqrt{0.1} end{bmatrix}$

is also a valid singular value decomposition.

### Other articles related to "example":

Example

Example may also refer to:

• Example (musician), a British musician
• Example, Florida rock band For Squirrels' sole major-label album, released in 1995
• example.com, example.net, example.org, example.edu and .example, domain names reserved for use in documentation as examples
• HMS Example (P165), an Archer-class patrol and training vessel of the British Royal Navy
• The Example, a 1634 play by James Shirley
• The Example (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson
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