Eigenvalues and eigenvectors

Directions a matrix leaves alone

A square matrix $A$ moves most vectors both in length and in direction. A few special directions are only stretched: $A\mathbf v=\lambda\mathbf v$ with $\mathbf v\neq\mathbf 0$. Then $\mathbf v$ is an eigenvector and the stretch factor $\lambda$ is its eigenvalue. If $\lambda=2$ the vector doubles, if $\lambda=-1$ it flips, if $\lambda=0$ it collapses to zero. Everything a matrix does to its eigenvectors is therefore one number.

To find the eigenvalues, note that $(A-\lambda I)\mathbf v=\mathbf 0$ has a non-zero solution only when $A-\lambda I$ is singular, so $\det(A-\lambda I)=0$. For a $2\times2$ matrix this is $\lambda^2-(\operatorname{tr}A)\lambda+\det A=0$. Then for each root solve $(A-\lambda I)\mathbf v=\mathbf 0$ for the direction. Two facts give quick checks: the eigenvalues add up to the trace and multiply to the determinant. A matrix is singular exactly when one eigenvalue is zero, and $A^k$ has eigenvalues $\lambda^k$ with the same eigenvectors.

• Eigenpair: $A\mathbf v=\lambda\mathbf v$ with $\mathbf v\neq\mathbf 0$; an eigenvector's direction is unchanged, its length is scaled by $\lambda$.
• Characteristic equation: $\det(A-\lambda I)=0$; for $2\times2$, $\lambda^2-\operatorname{tr}A\,\lambda+\det A=0$.
• Checks: eigenvalues sum to the trace and multiply to the determinant; $A^k$ has eigenvalues $\lambda^k$.

Lab · what's graded

• What an eigenpair is15 pts
• The larger eigenvalue20 pts
• Determinant from eigenvalues10 pts
• The characteristic equation15 pts
• Squaring a matrix15 pts
• The method in order10 pts
• Why finance cares15 pts