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 transpose of a matrix
Definition
For , the transpose of is the matrix defined by
Thus transposition interchanges rows and columns. The transpose of a zero-sized matrix is the unique matrix of the transposed zero-sized shape.
Depends on
Used by
- Cartan involution and k plus p for sl n r Example
- Compact and split real forms of sl two c Example
- Orthogonal and special orthogonal Lie groups Example
- SU(2) to SO(3) as a covering homomorphism Example
- The real symplectic matrix group Example
- For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface Proposition
- Transpose is linear and involutive, and (AB)^T=B^TA^T Proposition
- Every invertible real or complex square matrix has a unique factorisation A=QR with Q orthogonal or unitary and R upper triangular with positive real diagonal Theorem
- In dual bases, the matrix of T^* is the transpose of the matrix of T Theorem
- In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix Theorem
Dependency tree · two levels
6 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
- S. Axler, Linear Algebra Done Right, 4th ed., §3C, Exercises 14–15 (standard reference, not scraped)