Rayleigh Quotient Optimizer

Math Linear Algebra • Applications and Advanced Linear Algebra (capstone)

Written by STEM Calculators Team

Compute the Rayleigh quotient \(R(x)=\dfrac{x^{\mathsf T}Ax}{x^{\mathsf T}x}\), use it for eigenvalue bounds (symmetric case), and run an optimizer that targets the maximum and/or minimum Rayleigh value (i.e., \(\lambda_{\max}\) and \(\lambda_{\min}\)).

Matrix size

Optimize

Iterations

Tolerance

Display decimals

Matrix \(A\)

The quadratic form \(x^{\mathsf T}Ax\) depends only on the symmetric part \(A_s=\tfrac12(A+A^{\mathsf T})\). This tool uses \(A_s\) for \(R(x)\), eigenvalue bounds, and optimization when \(A\) is not symmetric.

Inputs accept -3.5, 2e-4, fractions 7/3, and constants pi, e.

Vector \(x\)

Initial vector used to evaluate \(R(x)\) and to start the optimizer.

Tip: \(x\neq 0\). The optimizer normalizes \(x\) at each step.

Ready

Results

Rayleigh quotient \(R(x)\)

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Eigenvalue bounds (from \(A_s\))

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Maximizer (approx)

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Minimizer (approx)

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Key objects

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Iteration log (truncated)

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Graph

2×2 only. Plots \\(R(\\theta)\\) for unit vectors \\(x(\\theta)=[\\cos\\theta,\\sin\\theta]\\) and marks current/iteration points. Zoom with wheel/trackpad, drag to pan, double-click to reset. Play animates the chosen mode.

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Step-by-step

Enter inputs and click “Calculate”.

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Theory

Rayleigh quotient definition

For a real matrix \(A\) and a nonzero vector \(x\in\mathbb{R}^n\), the Rayleigh quotient is \[ R(x)=\frac{x^{\mathsf T}Ax}{x^{\mathsf T}x}. \] It is scale-invariant: \(R(cx)=R(x)\) for any nonzero scalar \(c\), because both numerator and denominator scale by \(c^2\). This means it is natural to study \(R(x)\) on the unit sphere \(\lVert x\rVert=1\), where it simplifies to \(R(x)=x^{\mathsf T}Ax\).

Although \(A\) may be nonsymmetric, the quadratic form \(x^{\mathsf T}Ax\) depends only on the symmetric part \(A_s=\tfrac12(A+A^{\mathsf T})\), since \(x^{\mathsf T}(A-A^{\mathsf T})x=0\). Therefore, for real-valued Rayleigh quotient analysis and eigenvalue bounds, it is standard to work with symmetric matrices (or with \(A_s\)).

Eigenvalue bounds and why they matter

If \(A\) is symmetric, it has real eigenvalues \(\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n\) and an orthonormal eigenbasis. A fundamental fact is: \[ \lambda_{\min}\le R(x)\le \lambda_{\max}\quad\text{for all }x\ne 0, \] where \(\lambda_{\min}\) and \(\lambda_{\max}\) are the smallest and largest eigenvalues of \(A\). In other words, the Rayleigh quotient can never exceed the spectral extremes. Moreover, the extremes are attained: \(R(x)=\lambda_{\max}\) exactly when \(x\) is an eigenvector for \(\lambda_{\max}\), and similarly for \(\lambda_{\min}\).

This is why the Rayleigh quotient is used as an eigenvalue bound tool. If you can quickly evaluate \(R(x)\) for a meaningful direction \(x\) (for example, a direction suggested by physics, geometry, or a previous iteration), you instantly get a valid lower/upper bound on the spectrum. In numerical linear algebra, this idea appears in error estimation, preconditioning diagnostics, and in interpreting quadratic energy functions.

Optimization viewpoint and the power method connection

Maximizing \(R(x)\) over \(\lVert x\rVert=1\) is an optimization problem: \[ \max_{\lVert x\rVert=1} x^{\mathsf T}Ax. \] The optimizer is the dominant eigenvector and the optimal value is \(\lambda_{\max}\). Similarly, minimizing gives \(\lambda_{\min}\). One way to see why eigenvectors are special is via a constrained-gradient argument: on the unit sphere, stationary points of \(R(x)\) satisfy \(Ax=\lambda x\), i.e., they are eigenvectors, and their Rayleigh quotient equals the corresponding eigenvalue.

A classical iterative approach is the power method: \[ x_{k+1}=\frac{Ax_k}{\lVert Ax_k\rVert}. \] For symmetric matrices with a unique dominant eigenvalue and a good starting vector, \(x_k\) tends to the dominant eigenvector. The Rayleigh quotient \(R(x_k)\) typically increases toward \(\lambda_{\max}\). To approximate \(\lambda_{\min}\), one can apply the same idea to \(-A\): maximizing \(R_{-A}(x)\) corresponds to minimizing \(R_A(x)\), because \(R_{-A}(x)=-R_A(x)\).

In practice, convergence depends on eigenvalue gaps (how separated the largest eigenvalues are), the conditioning of eigenvectors, and the choice of starting vector. A useful diagnostic is the eigen-residual \(\lVert Ax-\lambda x\rVert\). When this residual is small (with \(\lambda=R(x)\)), the vector is close to an eigenvector.

Geometric intuition (2D)

In two dimensions, restricting to the unit circle lets you represent every unit vector as \(x(\theta)=[\cos\theta,\sin\theta]\). Then \(R(\theta)=x(\theta)^{\mathsf T}Ax(\theta)\) becomes a smooth periodic function of \(\theta\). The maximum and minimum values of this curve occur exactly at the eigenvector directions, providing an intuitive picture of why the Rayleigh quotient “finds” eigenvalues. This is also why the visualization in the calculator plots \(R(\theta)\) and animates iteration points moving toward extremizers.

Frequently Asked Questions

Why does the calculator use A_s=(A+A^T)/2?

Because x^T A x equals x^T A_s x for all real x. Rayleigh quotient bounds are stated for symmetric matrices, so A_s is the relevant part.

What do the bounds mean?

For symmetric A_s, the Rayleigh quotient always lies between λ_min and λ_max. The extremes occur at eigenvectors.

How does the optimizer relate to the power method?

The update x_{k+1}=normalize(A_s x_k) is the power method. The Rayleigh quotient along iterates tends to the dominant eigenvalue under suitable conditions.

How is the minimum eigenvalue approximated?

By running the same iteration on -A_s. Maximizing the Rayleigh quotient of -A_s corresponds to minimizing the Rayleigh quotient of A_s.