Eigenvalues and Eigenvectors

An eigenvector of a matrix $A$ is a nonzero vector $\mathbf{u}$ that $A$ stretches or flips without changing direction: $A\mathbf{u} = \lambda \mathbf{u}$. The scalar $\lambda$ is the eigenvalue corresponding to that eigenvector. Think of eigenvectors as the "special directions" in space — they are the axes along which the transformation $A$ acts purely as scaling.

To find eigenvalues, rearrange: $A\mathbf{u} = \lambda \mathbf{u}$ means $(A - \lambda I)\mathbf{u} = \mathbf{0}$. For a nonzero solution $\mathbf{u}$ to exist, the matrix $(A - \lambda I)$ must be singular, which requires $\det(A - \lambda I) = 0$. This equation in $\lambda$ is called the characteristic equation and its solutions are the eigenvalues.

For a $2 \times 2$ symmetric matrix $A = \begin{pmatrix}a & b\\b & d\end{pmatrix}$, the characteristic equation is $\lambda^2 - (a+d)\lambda + (ad - b^2) = 0$, giving eigenvalues $\lambda = \frac{(a+d) \pm \sqrt{(a-d)^2 + 4b^2}}{2}$. Both eigenvalues are always real.