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Finite configurations with equal Gram matrices have isometric linear spans

Statement

Let $n \geq 0$ , $x_{1}, \dots, x_{n} \in R^{d}$ , and $y_{1}, \dots, y_{n} \in R^{m}$ be finite lists with $⟨ x_{i}, x_{j} ⟩ = ⟨ y_{i}, y_{j} ⟩$ for every $1 \leq i, j \leq n$ , using the standard Euclidean inner products. The assignment $\sum_{i = 1}^{n} c_{i} x_{i} \mapsto \sum_{i = 1}^{n} c_{i} y_{i}$ is a well-defined linear isometry from $span {x_{i}}$ onto $span {y_{i}}$ ; when $n = 0$ both spans are zero spaces and the assignment is the unique zero map.

Statement references

Hastie, Tibshirani, and Friedman, ESL II, §18.5.2, printed pp. 670–672

Cameron, Linear and Nonlinear Dimensionality Reduction, §3, PDF pp. 4–6

Facts & Assumptions

Given: An integer $n \geq 0$ and lists $x_{1}, \dots, x_{n} \in R^{d}$ and $y_{1}, \dots, y_{n} \in R^{m}$ with equal pairwise inner products.

[F1]

The standard coordinate pairings on $R^{d}$ and $R^{m}$ are positive-definite inner products, with $⟨ u, v ⟩ = \sum_{k} u_{k} v_{k}$ (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

Proof

1.1givenF1algebra

For any coefficients $c_{1}, \dots, c_{n}$ , bilinearity and the equal Gram entries give $∥ \sum_{i} c_{i} x_{i} ∥_{2}^{2} = \sum_{i, j} c_{i} c_{j} ⟨ x_{i}, x_{j} ⟩ = \sum_{i, j} c_{i} c_{j} ⟨ y_{i}, y_{j} ⟩ = ∥ \sum_{i} c_{i} y_{i} ∥_{2}^{2}$ . If $\sum_{i} c_{i} x_{i} = 0$ , the left side is zero, so [F1] implies $\sum_{i} c_{i} y_{i} = 0$ . Applying this to the difference of two coefficient representations shows that the assignment in the statement is well defined.

2.1step 1.1F1algebra∎

For coefficient lists $c, d$ , the same bilinear expansion gives $⟨ \sum_{i} c_{i} x_{i}, \sum_{j} d_{j} x_{j} ⟩ = \sum_{i, j} c_{i} d_{j} ⟨ x_{i}, x_{j} ⟩ = \sum_{i, j} c_{i} d_{j} ⟨ y_{i}, y_{j} ⟩ = ⟨ \sum_{i} c_{i} y_{i}, \sum_{j} d_{j} y_{j} ⟩$ . The coefficient formula is linear, and its range contains every $y_{i}$ because it maps $x_{i}$ to $y_{i}$ ; hence its range is exactly their span and it preserves inner products. If $n = 0$ , both spans are zero spaces and this is their unique zero map. Thus in every case the assignment is an onto linear isometry.

Depends on

The standard formulas $\langle x,y\rangle=\sum_{k<n}x_k y_k$ on $\mathbb R^n$ and $\sum_{k<n}x_k\overline{y_k}$ on $\mathbb C^n$ are inner products

Used by

Double-centering Euclidean squared distances recovers their centered Gram matrix Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Elements of Statistical Learning, second edition, corrected 12th printing (standard reference, not scraped)
Maria Cameron, Linear and Nonlinear Dimensionality Reduction, lecture notes (standard reference, not scraped)