Read the equation
The matrix is I+φxxᵀ. For any nonzero vector v, vᵀgv=||v||²+φ(x·v)²>0, so it is positive definite for real finite x. It stretches lengths more strongly along x while leaving directions orthogonal to x unchanged; its eigenvalues are 1+φ||x||² along x and 1 in orthogonal directions.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| g | Metric matrix |
| x | Coordinate vector |
| delta_ij | Kronecker identity matrix |
| phi | Positive golden ratio |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
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Original context
Read the surrounding source passage
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// ═══════════════════════════════════════════════════════════════════════════════
// DIFFERENTIAL GEOMETRY EQUATIONS
// ═══════════════════════════════════════════════════════════════════════════════
// MetricTensor computes g_ij for a manifold point
// g_ij = δ_ij + φ·x_i·x_j (golden-scaled metric)
func MetricTensor(x []float64) [][]float64 {
n := len(x)
g := make([][]float64, n)
for i := range g {
g[i] = make([]float64, n)
for j := range g[i] {
if i == j {
g[i][j] = 1.0 + PHI*x[i]*x[j]
} else {
g[i][j] = PHI * x[i] * x[j]Exact original notation
g_ij = δ_ij + φ·x_i·x_j (golden-scaled metric)