Matrix sign function
In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function.
It was introduced by J.D. Roberts in 1971 as a tool for model reduction and for solving Lyapunov and Algebraic Riccati equation|Riccati] equation in a technical report of Cambridge University, which was later published in a journal in 1980.
Definition
The matrix sign function is a generalization of the complex signum functionto the matrix valued analogue. Although the sign function is not analytic, the matrix function is well defined for all matrices that have no eigenvalue on the imaginary axis, see for example the Jordan-form-based definition.
Properties
Theorem: Let, then.Theorem: Let, then is diagonalizable and has eigenvalues that are.
Theorem: Let, then is a projector onto the invariant subspace associated with the eigenvalues in the right-half plane, and analogously for and the left-half plane.
Theorem: Let, and be a Jordan decomposition such that corresponds to eigenvalues with positive real part and to eigenvalue with negative real part. Then, where and are identity matrices of sizes corresponding to and, respectively.
Computational methods
The function can be computed with generic methods for matrix functions, but there are also specialized methods.Newton iteration
The Newton iteration can be derived by observing that, which in terms of matrices can be written as, where we use the matrix [square root]. If we apply the Babylonian method to compute the square root of the matrix, that is, the iteration, and define the new iterate, we arrive at the iterationwhere typically. Convergence is global, and locally it is quadratic.
The Newton iteration uses the explicit inverse of the iterates.
Newton–Schulz iteration
To avoid the need of an explicit inverse used in the Newton iteration, the inverse can be approximated with one step of the Newton iteration for the inverse,, derived by Schulz in 1933. Substituting this approximation into the previous method, the new method becomesConvergence is quadratic, but only local.
Applications
Solutions of Sylvester equations
Theorem: Let and assume that and are stable, then the unique solution to the Sylvester equation,, is given by such thatProof sketch: The result follows from the similarity transform
since
due to the stability of and.
The theorem is, naturally, also applicable to the Lyapunov equation. However, due to the structure the Newton iteration simplifies to only involving inverses of and.
Solutions of algebraic Riccati equations
There is a similar result applicable to the algebraic Riccati equation,. Define asUnder the assumption that are Hermitian and there exists a unique stabilizing solution, in the sense that is stable, that solution is given by the over-determined, but consistent, linear system
Proof sketch: The similarity transform
and the stability of implies that
for some matrix.