Rayleigh Quotient

The Rayleigh quotient is a scalar value associated with a vector x and a square matrix A. It’s defined as:

$$R_A(x)=\frac{x^{T}Ax}{x^{T}x}$$

We can think of the Rayleigh quotient as a way to “measure” how much a matrix stretches a vector in the direction of that vector.

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If $A$ is symmetric (or Hermitian in complex space), the Rayleigh quotient has a few very cool properties:

• If $x$ is an eigenvector of $A$, then:

$$R_A(x)=\frac{x^{T}Ax}{x^{T}x}=\lambda$$

where $\lambda$ is the corresponding **eigenvalue**. So the Rayleigh quotient gives the eigenvalue when you're already on the eigenvector.

• The maximum and minimum values of the Rayleigh quotient over unit vectors are the largest and smallest eigenvalues of the matrix $A$

🔍 The Rayleigh quotient tells you how “eigen-like” a vector is and approximates the eigenvalue if it’s close to an eigenvector.

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For an optimization problem over the Rayleigh Quotient:

$$ma{x}_{\Vert x\Vert =1}{x}^{T}Ax$$

The optimal solution is $x=v_{max}={\lambda }_{max}(A)$.

$v_{max}^{T}A{v}_{max}=v_{max}^T{\lambda }_{max}{v}_{max}={\lambda }_{max}$ given that the eigenvectors are unit eigenvectors.