Alphabeta MathHow 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.
The non-negative square root of an explicit matrix is exhibited as a polynomial in the matrix
Example
For
the non-negative square root is
Facts & Assumptions
Given: The real symmetric matrix above.
A self-adjoint real operator has an orthonormal eigenbasis (Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis).
The non-negative square root of a non-negative operator is a polynomial in the operator (The non-negative square root of a non-negative operator is a polynomial in the operator).
Verification
One has and , so after normalising these vectors, [L1] gives an orthonormal eigenbasis with eigenvalues and . Therefore is non-negative.
In the same eigenbasis, the non-negative square root has eigenvalues and , which corresponds in the standard basis to . Direct multiplication gives , and direct algebra gives , exactly as [L2] predicts.
Depends on
- The non-negative square root of a non-negative operator is a polynomial in the operator
- Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis
- 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
Nothing in the library uses this result yet.
Dependency tree · two levels
12 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
- Sheldon Axler, Linear Algebra Done Right, fourth edition (standard reference, not scraped)