Example · Undergraduate
Terms
| Term | This example | Why |
|---|
| eigenvalue |
$2$ and $-3$
|
The eigenvalues are $2$ and $-3$, because \(\begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 0 \end{bmatrix} = 2 \begin{bmatrix} 1 \\ 0 \end{bmatrix}\) and \(\begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ -3 \end{bmatrix} = -3 \begin{bmatrix} 0 \\ 1 \end{bmatrix}\).
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| singular value |
$3$ and $2$
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The singular values of \(A = \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}\) are $3$ and $2$, the square roots of the eigenvalues $9$ and $4$ of \(A^{\top} A = \begin{bmatrix} 4 & 0 \\ 0 & 9 \end{bmatrix}\). $A$ is symmetric, so its singular values are the absolute values of its eigenvalues.
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Distinctions using this example
- Eigenvalue vs. singular value: The diagonal matrix with entries $2$ and $-3$ has eigenvalues $2$ and $-3$, its diagonal entries. Its singular values are their absolute values, $3$ and $2$, as for every symmetric matrix.
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