LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Symmetric matrices are Euclidean under the Frobenius norm

Statement

Let $d \geq 1$ , let $S^{d}$ be the real vector space of symmetric $d \times d$ matrices, and set $m = d (d + 1) / 2$ . Use one-based entry labels $A_{i j} : = A (i - 1, j - 1)$ for $1 \leq i, j \leq d$ , and likewise for every matrix below. Equip $S^{d}$ with the Frobenius inner product $⟨ A, B ⟩_{F} : = \sum_{i = 1}^{d} \sum_{j = 1}^{d} A_{i j} B_{i j}$ and its induced norm. Define $svec : S^{d} \to R^{m}$ by listing $A_{11}, \dots, A_{d d}$ first, followed by $\sqrt{2} A_{i j}$ for $1 \leq i < j \leq d$ in lexicographic order. Then $svec$ is a linear bijection and preserves inner products and norms; in particular, $∥ svec (A) - svec (B) ∥_{2} = ∥ A - B ∥_{F}$ for all $A, B \in S^{d}$ .

Statement references

Entry-sum definition of the Frobenius norm

Euclidean inner product on real coordinate space

Positive square root used in the off-diagonal coordinates

Higham, §2, Theorem 2.1, printed pp. 105–106

Bindel, §2, PDF pp. 1–2

Facts & Assumptions

Given: An integer $d \geq 1$ , symmetric real matrices $A, B \in S^{d}$ , and $m = d (d + 1) / 2$ .

[F1]

For a real matrix, the squared Frobenius norm is the sum of the squares of all entries (The Frobenius norm $∥ A ∥_{F} = (\sum_{i, j} ∣ a_{i j} ∣^{2})^{1 / 2}$ on real or complex matrices). Under the stated entry labels, substituting $i = r + 1$ , $j = s + 1$ gives $\sum_{r < d} \sum_{s < d} A (r, s)^{2} = \sum_{i = 1}^{d} \sum_{j = 1}^{d} A_{i j}^{2}$ ; in particular, when $d = 1$ , $A_{11} = A (0, 0)$ .

[F2]

The Euclidean inner product on $R^{m}$ is the coordinate dot product, with norm the square root of the sum of coordinate squares (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[F3]

Every nonnegative real number has a unique nonnegative square root, so $\sqrt{2} > 0$ and division by $\sqrt{2}$ is defined (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Proof

1.1givenF3algebra

The coordinate list has $d + d (d - 1) / 2 = m$ entries. Its first $d$ entries recover each diagonal entry of $A$ , and each later entry recovers the corresponding symmetric pair by division by $\sqrt{2}$ from [F3]. Thus it has a unique inverse on $R^{m}$ ; its coordinate formulas are linear, so $svec$ is a linear bijection.

1.2F2givenalgebra

For $A, B \in S^{d}$ , the Euclidean dot product of their coordinate lists is $\sum_{i} A_{i i} B_{i i} + 2 \sum_{i < j} A_{i j} B_{i j}$ , because each off-diagonal coordinate contributes $(\sqrt{2} A_{i j}) (\sqrt{2} B_{i j})$ . Symmetry gives $⟨ A, B ⟩_{F} = \sum_{i} A_{i i} B_{i i} + \sum_{i < j} (A_{i j} B_{i j} + A_{j i} B_{j i})$ , which is the same expression. Hence $⟨ svec (A), svec (B) ⟩_{2} = ⟨ A, B ⟩_{F}$ .

2.1step 1.2F1F2F3algebra∎

Linearity gives $svec (A) - svec (B) = svec (A - B)$ . Applying step 1.2 to $A - B$ with itself and taking the unique nonnegative square roots from [F1]–[F3] yields $∥ svec (A) - svec (B) ∥_{2} = ∥ A - B ∥_{F}$ . This proves norm preservation.

Depends on

Used by

Dependency tree · two levels

27 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

Nicholas J. Higham, Computing a Nearest Symmetric Positive Semidefinite Matrix, Linear Algebra and its Applications 103 (1988), 103–118 (standard reference, not scraped)
David Bindel, Matrix Computations, CS 6210, Matrix Nearness Problems, Fall 2022 (standard reference, not scraped)