Exercise 2213Linear algebra › Quadratic formsL1Added 3 Nov 2026

The Rayleigh quotient

#Rayleigh quotient#eigenvalues#optimization

Statement

Let $A$ be a symmetric matrix. The Rayleigh quotient of $X\neq0$ is
$R(X)=\frac{X^TAX}{X^TX}.$
Show that the minimum and maximum of $R$ over $\mathbb{R}^n\setminus\{0\}$ are the smallest and largest eigenvalues of $A$ .

Hint

Use the Lagrange multiplier method on the sphere, or the orthogonal diagonalization $A=QDQ^T$ .

Show solutionHide solution

Solution

Diagonalization. $A=QDQ^T$ with $Q$ orthogonal, $D=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ .

Change of variables. Let $Y=Q^TX$ . Then $X^TAX=Y^TDY=\sum_i\lambda_i y_i^2$ and $\|X\|^2=\|Y\|^2=\sum_i y_i^2$ . So
$R(X)=\frac{\sum_i\lambda_i y_i^2}{\sum_i y_i^2}.$

Bounding. Since $\lambda_1\le\lambda_i\le\lambda_n$ , the quotient lies in $[\lambda_1,\lambda_n]$ .

Attainment. Take $Y$ proportional to $e_1$ (for $\lambda_1$ ) or $e_n$ (for $\lambda_n$ ); then $X=QY$ is the corresponding eigenvector.

Conclusion. $\min R=\lambda_{\min}$ and $\max R=\lambda_{\max}$ .

The Rayleigh quotient

Statement

Solution

More problems in Quadratic forms