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43 results for “Eigenvalues” · Page 2 of 3
Optimization · Lesson
The Hessian's eigenvalues reveal the type: all positive = minimum, all negative = maximum, mixed = saddle.
Machine Learning · Lesson
Covariance & its Eigenvectors Drag points and watch the covariance, eigenvalues, and arrows update.
Aperiodic Tilings · Lesson
… matrix [[2, 1], [1, 1]] — trace 3, determinant 1, so its eigenvalues are (3 ± √5)/2 and the larger is φ².
Symmetry · Lesson
… leading eigenvector of the substitution matrix has component ratio φ (its leading eigenvalue is φ² = φ + 1), making the golden ratio an unavoidable feature of the tiling's combinatorial structure.
Differential Equations · Lesson
… outward, or orbit in closed loops -- is determined entirely by the eigenvalues of the coefficient matrix.
Differential Geometry · Lesson
Eigenvalues of the Shape Operator The principal curvatures are the eigenvalues of the shape operator S, and the principal directions are the corresponding eigenvectors .
Quantum · Module overview
See also: Linear Algebra for Hilbert spaces and operator eigenvalues, Fourier Analysis for the quantum Fourier transform behind Shor's algorithm, and Representation Theory for symmetry groups acting on quantum states.
Quantum · Lesson
Given a unitary operator U and its eigenstate, phase estimation finds the eigenvalue phase to n bits of precision using n counting qubits.
Quantum · Lesson
I² = I Trivial rotation (0°) Eigenvalues: 1, 1 Pauli-X (NOT) Bit flip: swaps |0⟩ and |1⟩.
Random Matrices · Module overview
Fill a large matrix with random numbers and its eigenvalues form a semicircle and refuse to touch.
Random Matrices · Lesson
… zeros of the Riemann zeta function come from pure arithmetic; random matrix eigenvalues come from pure probability.
Lie Groups · Lesson
They arise from the eigenvalues of the adjoint representation — the roots are the "weights" by which the algebra acts on itself.
Loop Quantum Gravity · Lesson
The area gap of Planck areas is an eigenvalue, and a boosted observer measures a different operator rather than a contracted version of the same one, exactly as the eigenvalues of angular momentum are the same integers in every frame with the rotation group …
Differential Equations · Lesson
The eigenvalues of J determine the local behavior.
Lie Groups · Lesson
… -2I => SO(3) is compact Semisimple: det(B) != 0 Has both +/- eigenvalues => SL(2,R) is non-compact Semisimple: det(B) != 0 Signature: (1,2) or (2,1) B = 0 (all entries) Nilpotent algebra NOT semisimple rgba(255, 255, 255, 0.4) 11px sans-serif Color key: rgb(230, 160, …
Sphere Packing · Lesson
… because each one is an eigenfunction of the n-dimensional Fourier transform with eigenvalue (−1) k , so f̂ comes out exactly rather than by numerical integration.
Loop Quantum Gravity · Lesson
… | MathAtlas The area of a surface becomes a self-adjoint operator whose eigenvalues are sums of square roots of j(j+1), with a smallest nonzero value of about 5.17 Planck areas, and a volume operator that lives at the nodes and is far harder.
Differential Geometry · Lesson
They are the eigenvalues of the shape operator, and their corresponding eigenvectors are the principal directions .
Random Matrices · Lesson
Fill a matrix with noise and its eigenvalues form a semicircle: measurable.
Random Matrices · Lesson
… matrix with bell-curve numbers, coin flips or uniform draws: the gaps between eigenvalues come out identical.