Spectral Matrix Sign
Split a symmetric matrix into positive and negative spectral subspaces.
Free theory articleSplit a symmetric matrix into positive and negative spectral subspaces.
Concept & interpretation
For a real symmetric matrix A = R diag(λ1, λ2) RT with no zero eigenvalue, the matrix sign function is defined as sign(A) = R diag(sign(λi)) RT. It acts spectrally on the invariant subspaces. If A has any zero eigenvalue, the matrix sign function is undefined. The projector onto the invariant subspace associated with the positive eigenvalues is P+ = (I + sign(A))/2.
Experiment
Open the Spectral Matrix Sign module to explore how the spectrum and basis angle influence the sign matrix.
- Set Spectrum to "Indefinite" and Angle to 45. Observe the Sign entries: the diagonal is 0 and the off-diagonals are -1 (or 1 depending on basis ordering).
- Set Angle to 0. The matrix sign is exactly diag(-1, 1) or diag(1, -1) matching the sign of the eigenvalues.
- Change Spectrum to "Positive". The Positive rank becomes 2, and sign(A) is the identity matrix.
- Change Spectrum to "Singular" or "Zero". Note that the status correctly reports "Undefined at zero eigenvalue" and no false involution is presented.