Euclidean distance matrix

A Euclidean distance matrix (EDM) is a real $m\\times m$ matrix $X$ such that for some points $y_{1},\\ldots,y_{m}$ in $\\mathbb{R}^{m}$ , $X_{ik}=\\left\\|y_{i}-y_{k}\\right\\|_{2}^{2}$ , where $\\|\\cdot\\|_{2}$ is the 2-norm on $\\mathbb{R}^{m}$ .

A EDM $X$ inherits the following from the norm that defines it:

•
$X_{ii}=0$ ;
•
$X_{ij}=X_{ji}\\geq 0$ ;
•
$\\sqrt{X_{ik}}\\leq\\sqrt{X_{ij}}+\\sqrt{X_{jk}}$ .

Additionally, $X$ is a EDM if and only if the diagonal entries of $X$ are all 0 and for all $z\\in\\mathbb{R}^{m}$ whose components sum to 0, $z^{T}Xz\\leq 0$ .

Finally, the set of $m\\times m$ EDMs forms a convex cone (http://planetmath.org/Cone3) in the set of all $m\\times m$ matrices.

References

1 S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.

单词	EuclideanDistanceMatrix
释义	Euclidean distance matrix A Euclidean distance matrix (EDM) is a real $m\\times m$ matrix $X$ such that for some points $y_{1},\\ldots,y_{m}$ in $\\mathbb{R}^{m}$ , $X_{ik}=\\left\\\|y_{i}-y_{k}\\right\\\|_{2}^{2}$ , where $\\\|\\cdot\\\|_{2}$ is the 2-norm on $\\mathbb{R}^{m}$ . A EDM $X$ inherits the following from the norm that defines it: • $X_{ii}=0$ ; • $X_{ij}=X_{ji}\\geq 0$ ; • $\\sqrt{X_{ik}}\\leq\\sqrt{X_{ij}}+\\sqrt{X_{jk}}$ . Additionally, $X$ is a EDM if and only if the diagonal entries of $X$ are all 0 and for all $z\\in\\mathbb{R}^{m}$ whose components sum to 0, $z^{T}Xz\\leq 0$ . Finally, the set of $m\\times m$ EDMs forms a convex cone (http://planetmath.org/Cone3) in the set of all $m\\times m$ matrices. References 1 S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
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