Finding the associated unit eigenvector

Mathematics Asked by Laufen on September 30, 2020


Find the eigenvalues $λ_1<λ_2$ and two associated unit eigenvectors of the symmetric matrix

$$A = begin{bmatrix}-7&12\12&11end{bmatrix}$$

My work so far

$$A = begin{bmatrix}-7-λ&12\12&11-λend{bmatrix}=λ^2-4λ-221=(λ+13)(λ-17)$$



To find the solution set for $λ_1$


and the solution set for $λ_2$




However, I’m unsure how to get the associated unit eigenvectors. Would I plug these into the quadratic formula to find the solutions? For example, for $λ_2$


I know that I’m off here, but just took a guess.

One Answer

As you correctly found for $lambda_{1}=-13$ the eigenspace is $(−2x_{2},x_{2})$ with $x_{2}inmathbb{R}$. So if you want the unit eigenvector just solve:

$(−2x_{2})^2+x_{2}^2=1^2$, which geometrically is the intersection of the eigenspace with the unit circle.

Correct answer by Gio on September 30, 2020

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