Find the principal angle between the subspace spanned by (1, 0, 0) and (0, 1, 0) and the line spanned by (1, 1, 1) in R^3. Give the angle in degrees.

Find the principal angle between the subspace spanned by (1, 0, 0) and (0, 1, 0) and the line spanned by (1, 1, 1) in R^3. Give the angle in degrees.

Approach: Project the line's direction vector onto the plane with the orthogonal projection matrix, then take the cosine of the angle as the ratio of the projected length to the original length.

35.26 degrees. The first subspace is the x-y plane, whose orthogonal projection sends v = (1, 1, 1) to Pv = (1, 1, 0). The principal angle satisfies cos(theta) = ||Pv|| / ||v|| = sqrt(2)/sqrt(3) = sqrt(2/3) = 0.8165, so theta = arccos(0.8165) = 35.26 degrees. Only one angle exists here because the smaller subspace has dimension 1. In general the cosines of the principal angles between two subspaces are the singular values of Q_1^T Q_2 for orthonormal bases Q_1 and Q_2, and this case is the single singular value 0.8165.

Follow-up: Two 3 dimensional factor spaces estimated on consecutive years give principal angle cosines 0.99, 0.95 and 0.20. What does that pattern say about the stability of the factors?

Key concepts: principal angle, orthogonal projection, subspace, cosine.