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The Gram matrix and Gram determinant, with empty value
Definition
For a finite list in a real or complex inner product space (Real and complex inner product spaces, with the inner product linear in the first argument), its Gram matrix is
using the square matrix space The vector space of by matrices over a field, with entrywise operations. Its Gram determinant is the determinant (For , the determinant over a commutative ring by the Leibniz formula, and for a real matrix) . For the empty list, the Gram matrix is the unique matrix and its determinant is .
Depends on
- Real and complex inner product spaces, with the inner product linear in the first argument
- The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations
- For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix
Used by
- Polar integration may discard the cut locus Corollary
- Surface integration on compact C1 hypersurfaces Definition
- The first fundamental form, Gram matrix, and area density of a surface patch Definition
- Projection onto a finite-dimensional subspace by a Gram matrix Example
- A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent Theorem
- The Gram formula gives a well-defined positive-definite inner product on exterior powers, and ‖v₁∧⋯∧ vₖ‖² is the Gram determinant Theorem
- The squared cross-product norm is the Gram determinant of two vectors Theorem
Dependency tree · two levels
19 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
- Kenneth Hoffman and Ray Kunze, Linear Algebra, 2nd ed., p. 332, Theorem 7 (standard reference, not scraped)