Construction of an eigenbasis of a symmetric matrix

Symmetric matrices are diagonalizable. This video animates the process of finding an eigenbasis.

00:00 The matrix defines an ellipsoid.
00:08 The first eigenvector (red) is the vector of largest extension of the ellipsoid.
00:16 The remaining eigenvectors are orthogonal to the first one.
00:24 Repeat the process in the orthogonal complement of the first eigenvector: the second (green) eigenvector has the direction of the largest dimension of the ellipse, the third (blue) is orthogonal to that.
01:04 Back in the original coordinate system, we have three vectors that do not change direction under the action of the matrix, they are eigenvectors.