Spectral Matrix Sign
Split a symmetric matrix into positive and negative spectral subspaces.
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Spectral Matrix Sign
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.
Worked Example
Consider an indefinite spectrum with eigenvalues λ1 = -2, λ2 = 3, and an angle of 45°.
- At 45°, the rotation matrix R is [[1/√2, -1/√2], [1/√2, 1/√2]].
- Since the eigenvalues are non-zero, sign(A) is defined. sign(λ1) = -1, sign(λ2) = 1.
- sign(A) = R diag(-1, 1) RT = [[0, -1], [-1, 0]].
- Verify the involution property: sign(A)2 = [[0, -1], [-1, 0]] [[0, -1], [-1, 0]] = [[1, 0], [0, 1]] = I.
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.
Limitations
The matrix sign function is strictly undefined when any eigenvalue has a zero real part, which for real symmetric matrices implies a zero eigenvalue. Some mathematical software silently return 0 for sign(0), but this breaks the fundamental involution property (sign(A)2 = I) and the algebraic relationship to spectral projectors. This module explicitly enforces the undefined boundary case. The interactive visualization is restricted to a 2x2 real symmetric domain. In general numerical practice for large matrices, sign(A) is often computed via iterative methods (such as Newton's method: Xk+1 = (Xk + Xk-1)/2) rather than explicit diagonalization.
References
The following reference describes the properties and computation of the matrix sign function:
- What Is the Matrix Sign Function? by Nicholas J. Higham. This article defines the sign function, its relation to spectral projectors, and emphasizes that it is undefined for matrices with purely imaginary eigenvalues (including zero).
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Spectral Matrix Sign module.
Try Spectral Matrix Sign on Riano →