A Few Matrix Applications

Matrix Eigenvalues. Eigenvalues are special scalars that describe how a matrix transforms space. "Eigen" means "its own" in German, meaning, the matrix's own value. Given a square matrix $A$, an eigenvalue $\lambda$ is a number such that there exists a non-zero vector $v$ satisfying:

$Av = \lambda v$

This means that multiplying $v$ by $A$ only stretches or shrinks it (by $\lambda$), without changing its direction.

To compute eigenvalues, we solve the characteristic equation:

$\det(A - \lambda I) = 0$

This produces a polynomial in $\lambda$, whose roots are the eigenvalues.