The Spectral Theorem: Symmetric Means Perpendicular Eigenvectors

Here are the three eigenvector directions of a three by three matrix, drawn as lines through the origin, with the unit sphere around them.

When the matrix is symmetric, the spectral theorem says three things: its eigenvalues are real, its eigenvectors are perpendicular, and A equals Q lambda Q transposed with Q orthonormal.

The rest of the narration is in the book's Companion.