Rayleigh quotient

• categories: Linear algebra, Definition

Definition

The Rayleigh quotient of a square matrix $A\in {\mathbb{R}}^{n\times n}$ and a nonzero vector $\mathbf{x}\in {\mathbb{R}}^n$ is defined as:

$$R(\mathbf{x},A)=\frac{{\mathbf{x}}^{T}A\mathbf{x}}{{\mathbf{x}}^T\mathbf{x}}.$$

• Numerator: ${\mathbf{x}}^{T}A\mathbf{x}$ is a scalar obtained by a quadratic form of $A$ and $\mathbf{x}$.
• Denominator: ${\mathbf{x}}^T\mathbf{x}=\Vert \mathbf{x}{\Vert }_2^2$ is the squared Euclidean norm of $\mathbf{x}$, ensuring normalization.

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Key Properties

1. Eigenvalues and eigenvectors:

   • If $\mathbf{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$, then: $$R(\mathbf{v},A)=\lambda .$$
   • The Rayleigh quotient evaluated at an eigenvector returns its corresponding eigenvalue.

2. Bounds on Eigenvalues:
   The Rayleigh quotient provides bounds for the eigenvalues of $A$. If $A$ is symmetric:

   $${\lambda }_{\min }(A)\le R(\mathbf{x},A)\le {\lambda }_{\max }(A),$$

   where ${\lambda }_{\min }$ and ${\lambda }_{\max }$ are the smallest and largest eigenvalues of $A$, respectively.

3. Maximization:
   For Symmetric Matrix $A$, the maximum value of $R(\mathbf{x},A)$ is the largest eigenvalue, achieved when $\mathbf{x}$ is the corresponding eigenvector:

   $$\underset{\mathbf{x}\ne 0}{\max }R(\mathbf{x},A)={\lambda }_{\max }.$$

4. Minimization:
   Similarly, the minimum value of $R(\mathbf{x},A)$ is the smallest eigenvalue, achieved when $\mathbf{x}$ is the corresponding eigenvector:

   $$\underset{\mathbf{x}\ne 0}{\min }R(\mathbf{x},A)={\lambda }_{\min }.$$

5. Symmetry and Definiteness:

   • For symmetric positive definite matrices, $R(\mathbf{x},A)>0$ for all $\mathbf{x}\ne 0$.
   • For symmetric negative definite matrices, $R(\mathbf{x},A)<0$ for all $\mathbf{x}\ne 0$.