## Sequence Converges to the Largest Eigenvalue of a Matrix

## Problem 403

Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$.

Furthermore, suppose that

\[|\lambda_1| > |\lambda_2| \geq \cdots \geq |\lambda_n|.\]
Let

\[\mathbf{x}_0=c_1\mathbf{u}_1+c_2\mathbf{u}_2+\cdots+c_n\mathbf{u}_n\]
for some real numbers $c_1, c_2, \dots, c_n$ and $c_1\neq 0$.

Define

\[\mathbf{x}_{k+1}=A\mathbf{x}_k \text{ for } k=0, 1, 2,\dots\]
and let

\[\beta_k=\frac{\mathbf{x}_k\cdot \mathbf{x}_{k+1}}{\mathbf{x}_k \cdot \mathbf{x}_k}=\frac{\mathbf{x}_k^{\trans} \mathbf{x}_{k+1}}{\mathbf{x}_k^{\trans} \mathbf{x}_k}.\]

Prove that

\[\lim_{k\to \infty} \beta_k=\lambda_1.\]