Gram matrix

[[concept]]

[!themes] Topics

Evaluation Error: SyntaxError: Unexpected token '>'

at DataviewInlineApi.eval (plugin:dataview:19027:21)

at evalInContext (plugin:dataview:19028:7)

at asyncEvalInContext (plugin:dataview:19038:32)

at DataviewJSRenderer.render (plugin:dataview:19064:19)

at DataviewJSRenderer.onload (plugin:dataview:18606:14)

at DataviewJSRenderer.load (app://obsidian.md/app.js:1:1182416)

at DataviewApi.executeJs (plugin:dataview:19607:18)

at DataviewCompiler.eval (plugin:digitalgarden:10763:23)

at Generator.next (<anonymous>)

at eval (plugin:digitalgarden:90:61)

Gram Matrix

Let $v_{1}, \dots, v_{n} \in F^{k}$ be a (finite) collection of vectors in $F^{k}$ , which has inner product $⟨ \cdot, \cdot^{'} ⟩$ . The Gram Matrix $G$ of the $v_{i}$ is given by

G_{i j} = ⟨ v_{i}, v_{j} ⟩, G = V^{*} V

Where $V$ is the matrix with $v_{i}$ as its $i$ th column

Notes

The lengths of the $v_{i}$ are the entries along the diagonal
The angles between each pair are on the off-diagonal

This information describes the entire geometry of the set of vectors; If any other set of vectors $u_{1}, \dots, u_{n}$ has the same Gram matrix, then there must exist some $Q \in O (k)$ such that $Q v_{i} = u_{i}$ for all $i$ .

Important

In this case, it is possible to get from one point cloud to the other via some orthogonal transformation - this is an if and only if!

References

Wikipedia

const modules = await cJS()

const COLUMNS = [  
	{ id: "Name", value: page => page.$link },  
	{ id: "Last Modified", value: page => modules.dateTime.getLastMod(page) },
];  
  
return function View() {  
	const current = dc.useCurrentFile();
// Selecting `#game` pages, for example. 
	let queryString = `@page and linksto(${current.$link})`;
	let pages = dc.useQuery(queryString);
	
	// check types
	pages = pages.filter( (p) => !modules.typeCheck.checkAll(p, current) ).sort()
	
	
	return <dc.Table columns={COLUMNS} rows={pages} paging={20}/>;  
}

const { dateTime } = await cJS()

return function View() {
	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}

Gram matrix

References

See Also

Mentions