Eigenvectors, eigenvalues, and eigenspaces

3b1b’s eigenvalue trick

This trick was put forth by 3blue1brown in this video for calculating the eigenvalues of any $2 \times 2$ matrix quickly. Since any $2 \times 2$ matrix has the properties

d e t (𝐴) = 𝜆 1 𝜆 2 T r (𝐴) = 𝜆 1 + 𝜆 2

if we take $𝑝 = d e t (𝐴)$ and $𝑚 = 12 T r (𝐴)$ , the eigenvalues are

𝜆 = 𝑚 \pm \sqrt 𝑚 2 - 𝑝

This last formula can easily be remembered with this jingle.

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3b1b’s eigenvalue trick

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