Rayleigh Quotient

The Rayleigh quotient is a fundamental scalar functional in linear algebra, spectral theory, and quantum mechanics that assigns a scalar energy value to any non-zero vector with respect to a Hermitian or real symmetric matrix. It connects the quadratic form of an operator to its underlying eigensystem, providing the mathematical foundation for variational optimization and cubic-order numerical eigensolvers.

Interactive experiment: exploring stationary points and bounds

Open the Rayleigh Quotient module. The module provides sliders for Matrix Entry a11, Matrix Entry a22, Off-diagonal Entry a12/a21, Vector Angle (theta), and Iteration Steps (k).

1. Set Matrix Entry a11 to 3, Matrix Entry a22 to 3, and Off-diagonal Entry a12/a21 to 1. Confirm that the calculated eigenvalue bounds display λ_max = 4.00 and λ_min = 2.00.
2. Adjust Vector Angle (theta) to 45°. Observe that the Rayleigh quotient readout reaches exactly 4.00, and the status indicator reports "Stationary (Eigenvector)". Notice that the mapped vector Ax points in the exact same direction as x with magnitude scaling 4.0.
3. Rotate Vector Angle (theta) to 135°. Observe that the Rayleigh quotient drops to 2.00, confirming the lower Courant-Fischer bound. Ax is once again collinear with x, scaled by λ = 2.0.
4. Set Vector Angle (theta) to 0° (where R(A, x) = 3.00). Now increase Iteration Steps (k) from 0 to 1. Observe that the RQI vector jumps directly to 45° with quotient 4.00, demonstrating rapid local convergence.
5. Test the boundary zero matrix by setting a11 = 0, a22 = 0, and a12/a21 = 0. Verify that for any angle (e.g. θ = 60°), R(A, x) evaluates cleanly to 0.00 without numerical overflow or divide-by-zero errors.