MIT OpenCourseWare
47 minutes
Eigenvalues of Symmetric, Skew-Symmetric, and Orthogonal Matrices in Linear Algebra
This concept compares the eigenvalues of three special kinds of matrices. A symmetric matrix ($A^T = A$) has only real eigenvalues. Its eigenvectors can always be chosen perpendicular to each other, so $A = Q\Lambda Q^T$. A skew-symmetric matrix ($A^T = -A$) has eigenvalues that are purely imaginary or zero. They sit on the imaginary axis of the complex plane. An orthogonal matrix ($Q^T Q = I$) has eigenvalues of absolute value 1. They sit on the unit circle. A symmetric matrix is positive definite exactly when all of its eigenvalues are greater than zero. Similar matrices ($B = M^{-1} A M$) always share the same eigenvalues. The singular value decomposition $A = U \Sigma V^T$ works for any matrix. Its singular values are the square roots of the eigenvalues of $A^T A$. These facts also predict how solutions of $\frac{d\mathbf{x}}{dt} = A\mathbf{x}$ behave. The real part of each eigenvalue controls growth or decay. The imaginary part controls oscillation.