[Math] Finding approximation of largest eigenvalue

Question

Given the following matrix, find an approximation of the largest eigenvalue.
$$ A =
\begin{bmatrix}
3 & 2 \\
7 & 5 \\
\end{bmatrix}
$$

And I was also given $$\vec x=
\begin{bmatrix}
1 \\
0 \\
\end{bmatrix}
$$

How my professor solves this is by calculating the slopes of $A\vec x = \vec b_1$, $A^2 \vec x = \vec b_2$, $A^3 \vec x = \vec b_3$ and so on until we get the slope of $\vec b_i$ converging to the same value. Then when we get the approximated $\vec b$, he plug into $A \vec b = \lambda \vec b$, and the corresponding $\lambda$ is the largest eigenvalue.

Since slope is $\frac yx$ , it works fine for $2 x 2$ matrix. But how do I apply this method for a bigger matrix?

Answer 1

What you mention, is a known numerical analysis method for the approximation of the largest (by absolute value) eigenvalue of a matrix.

Let $A\in \mathbb R^{n\times n}$ have $n$ linearly independent eigenvalues $\{ u \}_{i=1}^n$ as well as a unique eigenvalue $λ_1$ such that : $|λ_1| < |λ_2| \leq \dots \leq |λ_n|$ where $λ_1 \in \mathbb R$ and $u_1 \in \mathbb R^n$ : $Au_1=λ_1u_1.$

The method "Power Iteration" :

$$\begin{cases} x_k= Ax_{k-1} \to x_k = A^kx_0 \quad k=1,2,\dots \\ x_0 \end{cases}$$

Thorem : The method "Power Iteration" converges $\forall x_0$ (that is adequate for the problem given) and it is :

$$\lim_{k\to \infty} ε_k\frac{x_k}{||x_k||_2}=u_1$$

$$\lim_{k \to \infty} \frac{x_{k,i}}{x_{k-1,i}}=λ_1 \quad \forall \space i=1,2,\dots,n \quad \text{with} \quad u_{1,i} \neq 0 $$

where $\{ ε_k\}_{k=1}^\infty = \{ \pm1\}$ and $u_1$ eigenvector of $A$ with $||u_1||_2=1$.

I've given you a formal explanation of the method according to my old notes and my knowledge, for more, check here.