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## Homework Statement

Use Lagrange multipliers to find the eigenvalues and eigenvectors of the matrix

[itex]A=\begin{bmatrix}2 & 4\\4 & 8\end{bmatrix}[/itex]

## Homework Equations

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## The Attempt at a Solution

The book deals with this as an exercise. From what I understand, it says to consider the function [itex]f(x,y) = \frac{1}{2}(A[x,y]) \cdot [x,y][/itex], with the assumption that [itex]A[/itex] is symmetric (which is the case here).

It then asks what the gradient of the function is, which is [itex]\nabla f(x,y) = A[x,y][/itex].

It then asks to restrict [itex]f[/itex] to the region [itex]S=\{[x,y] \in \mathbb{R}^2 : x^2 + y^2 \leq 1\}[/itex]. Then, it then states that there must exists a vector [itex][x,y] \in S[/itex] and a real number [itex]\lambda \neq 0[/itex] such that [itex]A[x,y] = \lambda [x,y][/itex], and claiming that finding the maxima and minima of [itex]f[/itex] constrained to [itex]S[/itex] will give the eigenvalues and eigenvectors of [itex]A[/itex].

Why is this true? To be more precise, why do the eigenvalues and eigenvectors only exists in the unit disk? And, how do we know there exists such a [itex]\lambda \neq 0[/itex] that [itex]A[x,y] = \lambda [x,y][/itex]? I tried Lagrange multipliers, but I could only verify that [itex]A[x,y] = 2 \lambda [x,y][/itex].

Thanks.