= Solution
The <power method> starts with a unit vector $\mathbf x_0$ and iterates
$$
\mathbf x_{k+1}
=\frac{A\mathbf x_k}{\|A\mathbf x_k\|}.
$$
Its eigenvalue estimate is the <Rayleigh quotient>
$$
r(\mathbf x_k)
=\frac{\mathbf x_k^TA\mathbf x_k}
{\mathbf x_k^T\mathbf x_k}.
$$
By the <real spectral theorem>, choose an orthonormal eigenbasis $\mathbf v_1,\ldots,\mathbf v_n$ and write
$$
\mathbf x_0=\sum_{j=1}^nc_j\mathbf v_j.
$$
Assume $|\lambda_1|>|\lambda_2|$, $\lambda_1\ne0$, and $c_1\ne0$. Before normalization,
$$
A^k\mathbf x_0
=c_1\lambda_1^k
\left[
\mathbf v_1
+\sum_{j=2}^n
\frac{c_j}{c_1}
\left(\frac{\lambda_j}{\lambda_1}\right)^k
\mathbf v_j
\right].
$$
Thus the direction error is $O(|\lambda_2/\lambda_1|^k)$. Orthogonality makes the linear terms vanish from the Rayleigh quotient:
$$
r(\mathbf x_k)-\lambda_1
=
\frac{
\sum_{j=2}^nc_j^2\lambda_j^{2k}(\lambda_j-\lambda_1)
}{
\sum_{j=1}^nc_j^2\lambda_j^{2k}
}.
$$
Consequently
$$
\boxed{
r(\mathbf x_k)-\lambda_1
=O\!\left(\left|\frac{\lambda_2}{\lambda_1}\right|^{2k}\right)
}.
$$
This is the <quadratic Rayleigh-quotient improvement for the power method>.
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