2×2 eigenvalues

Japanese school year: University year 1

What you learn

Learn how to calculate the eigenvalues of a 2x2 matrix by solving its characteristic equation. An eigenvalue represents the scalar factor by which a nonzero vector is scaled under a linear transformation. Prior understanding of determinants and quadratic equation solving methods is required.

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Key points

This is the characteristic equation used to find eigenvalues $\lambda$ of matrix $A$ . Setting this determinant to 0 ensures that non-zero eigenvectors exist.

\det(A - \lambda I) = 0

This quadratic equation lets you quickly find eigenvalues $\lambda$ for a $2 \times 2$ matrix. Just plug in the diagonal sum $\operatorname{tr}(A)$ and determinant $\det(A)$ , then solve for $\lambda$ .

\lambda^2 - \operatorname{tr}(A)\lambda + \det(A) = 0

Use these relations to check if your calculated eigenvalues $\lambda_1$ and $\lambda_2$ are correct. Their sum equals the diagonal sum $\operatorname{tr}(A)$ , and their product equals $\det(A)$ .

\lambda_1 + \lambda_2 = \operatorname{tr}(A), \quad \lambda_1 \lambda_2 = \det(A)

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Practice 1Coming soon