📂Matrix Algebra

Rayleigh quotient

Definition

For a symmetric matrix $A$ and a vector $\mathbf{x}$, the following value is called the Rayleigh quotient.

$$ R(A, \mathbf{x}) = \dfrac{ \mathbf{x}^{\mathsf{T}} A \mathbf{x}}{\mathbf{x}^{\mathsf{T}} \mathbf{x}} $$

For complex matrices, for a Hermitian matrix $A$ the Rayleigh quotient is defined as follows.

$$ R(A, \mathbf{x}) = \dfrac{ \mathbf{x}^{\ast} A \mathbf{x}}{\mathbf{x}^{\ast} \mathbf{x}} $$

Explanation

If $\lambda$ is an eigenvalue of $A$ and $\mathbf{x}$ is an eigenvector corresponding to $\lambda$, then the Rayleigh quotient equals that eigenvalue.

$$ R(A, \mathbf{x}) = \lambda $$

If $\mathbf{x}_{k}$ converges to the dominant eigenvector, the Rayleigh quotient converges to the dominant eigenvalue. This computation is called the power method.

Properties

(a) The maximum (minimum) exists for the Rayleigh quotient, and its value is the largest (smallest) eigenvalue. $$ \lambda_{\min} \le R(A, \mathbf{x}) \le \lambda_{\max} $$

(b) For any scalar $c$, $R(A, \mathbf{x}) = R(A, c \mathbf{x})$.