Symmetric matrices (A = A^{\mathsf{T}}) are the best-behaved matrices in all of linear algebra, and the reason is a theorem of startling tidiness — the spectral theorem. It promises that every symmetric matrix has:

• real eigenvalues — never complex, and
• orthogonal eigenvectors — its eigen-directions are perpendicular.

So a symmetric matrix can always be diagonalized by a set of perpendicular axes — a pure stretch along a clean right-angled frame, with no skew anywhere.

Perpendicular, or not

Compare the two eigen-frames. The symmetric matrix's eigenvectors meet at a perfect right angle; the lopsided (non-symmetric) one's do not. Toggle between them and read the angle between the eigen-directions — exactly 90^\circ only when the matrix is symmetric.

The gateway to data science

Symmetric matrices aren't a curiosity — they're everywhere the same quantity meets itself. A covariance matrix, which records how the features of a dataset vary together, is always symmetric, so its eigenvectors are guaranteed perpendicular. Those perpendicular eigen-directions, ranked by eigenvalue, are precisely the axes that principal component analysis discovers. The spectral theorem is the quiet guarantee that makes PCA work at all.