Singular Value Decomposition/Examples/Arbitrary Example 1

Example of Singular Value Decomposition

Let $\mathbf A$ be the matrix specified as:

$\mathbf A := \begin {pmatrix} 2/3 & 0 \\ 5/6 & 1/2 \\ 1/3 & 1 \end {pmatrix}$

Then $\mathbf A$ has the following singular value decomposition:

$\begin {pmatrix} 2/3 & 0 \\ 5/6 & 1/2 \\ 1/3 & 1 \end {pmatrix} = \begin {pmatrix} 1/3 & 2/3 & 2/3 \\ 2/3 & 1/3 & -2/3 \\ 2/3 & -2/3 & 1/3 \end {pmatrix} \begin {pmatrix} \sqrt 2 & 0 \\ 0 & 1 / \sqrt 2 \\ 0 & 0 \end {pmatrix} \begin {pmatrix} 1 / \sqrt 2 & 1 / \sqrt 2 \\ 1 / \sqrt 2 & -1 / \sqrt 2 \end {pmatrix}$

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): singular value decomposition